Can you make money betting on random coin flips?
Can you make money betting on random coin flips?
If I offered you an investment opportunity which consisted of the chance to wager on a series of coin tosses, each with a 50/50 chance of doubling or halving your stake, would you take it? Probably not. If you were to invest a "lump sum" at the beginning and then just allow it to compound with each coin toss you'd end up with the sum of money you began with, no matter what the sequence of coin tosses, if there are an equal number of heads and tails (which becomes more probable as the number of coin tosses increases). Your net investment return would be zero, so it's not much of an investment opportunity. For example:
Beginning Stake = $100
Flip 1: Win = $200
Flip 2: Lose = $100
Flip 3: Lose = $50
Flip 4: Win = $100
So, this investment turns out to be a complete game of chance that you can only win if you are "lucky" and have more winning than losing tosses, and counting on luck is not a good investment strategy.
But is there a way to predictably make money on such an investment? Why yes, there is. If, instead of betting all your money on each coin toss, you wager half your money and keep the other half in "cash," you'll slowly get richer. For example:
Starting Value = $100
Flip 1: Win = $150 (double your $50 stake + the $50 not wagered)
Flip 2: Lose = $112.50
Flip 3: Lose = $84.37
Flip 4: Win = $126.56
A nice 26% return!
This phenomenon was first demonstrated by Claude Shannon, and referred to as Shannon's Demon. It shows it is possible to make money from an "asset" with random outcomes (coin tossing) that has a zero expected arithmetic geometric return. So, for example, it might well be possible to increase real portfolio returns by holding an asset such as gold that has a zero expected real return. It's just math.
Beginning Stake = $100
Flip 1: Win = $200
Flip 2: Lose = $100
Flip 3: Lose = $50
Flip 4: Win = $100
So, this investment turns out to be a complete game of chance that you can only win if you are "lucky" and have more winning than losing tosses, and counting on luck is not a good investment strategy.
But is there a way to predictably make money on such an investment? Why yes, there is. If, instead of betting all your money on each coin toss, you wager half your money and keep the other half in "cash," you'll slowly get richer. For example:
Starting Value = $100
Flip 1: Win = $150 (double your $50 stake + the $50 not wagered)
Flip 2: Lose = $112.50
Flip 3: Lose = $84.37
Flip 4: Win = $126.56
A nice 26% return!
This phenomenon was first demonstrated by Claude Shannon, and referred to as Shannon's Demon. It shows it is possible to make money from an "asset" with random outcomes (coin tossing) that has a zero expected arithmetic geometric return. So, for example, it might well be possible to increase real portfolio returns by holding an asset such as gold that has a zero expected real return. It's just math.
Last edited by Browser on Mon Jan 05, 2015 8:50 am, edited 1 time in total.
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Re: Can you make money betting on random coin flips?
It's a zero expected geometric return. The arithmetic expected return is .5(100%) + .5(50%) = 25%.Browser wrote:It shows it is possible to make money from an "asset" with random outcomes (coin tossing) that has a zero arithmetic expected return.
Re: Can you make money betting on random coin flips?
Already pointed out, it is a zero expected geometric return, but positive arithmetic real return.Browser wrote:It shows it is possible to make money from an "asset" with random outcomes (coin tossing) that has a zero arithmetic expected return. So, for example, it might well be possible to increase real portfolio returns by holding an asset such as gold that has a zero expected real return. It's just math.
This doesn't imply that gold could be a good investment even though it has zero expected real return. What matters it how it mixes with other assets in the portfolio. Bonds recently had negative expected real returns, but everyone still invested in them, right? (I say recently because now the breakeven inflation rate for 7year Treasuries and TIPS is only 1.55%). Our investments earn nominal dollars and we spend nominal dollars.
Re: Can you make money betting on random coin flips?
I find it interesting that when I Google "Shannon's Demon", I don't find a math site or a Wikipedia article, just a bunch of links to investing or trading web sites, including this Boglehead's discussion from 2013: http://www.bogleheads.org/forum/viewtop ... 0&t=116876.
Last edited by rkhusky on Mon Jan 05, 2015 9:57 am, edited 1 time in total.
Re: Can you make money betting on random coin flips?
....and BTW, the number of heads equaling the number of tails does not become "more probable as the number of coin tosses increases". The ratio of heads to tails converges to 1 as the number of tosses increases, but the difference in the number of heads to tails diverges to infinity as the number of coin tosses approaches infinity  one reason, aside from others, that you shouldn't hang out at the casino until you win you money back!Browser wrote: If you were to invest a "lump sum" at the beginning and then just allow it to compound with each coin toss you'd end up with the sum of money you began with, no matter what the sequence of coin tosses, if there are an equal number of heads and tails (which becomes more probable as the number of coin tosses increases). Your net investment return would be zero, so it's not much of an investment opportunity.
MB

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Re: Can you make money betting on random coin flips?
Yes, aware of this.
I do something like this in our retirement accounts to recover from 2008. We have recovered and hit the FI number. I think I need just one more year to be more than "safe, " but this one more year is going to be, The Gamble.
I do something like this in our retirement accounts to recover from 2008. We have recovered and hit the FI number. I think I need just one more year to be more than "safe, " but this one more year is going to be, The Gamble.
Rev012718; 4 Incm stream buckets: SS+pension; dfr'd GLWB VA & FI anntys, by time & $$ laddered; Discretionary; Rentals. LTCi. Own, not asset. Tax TBT%. Early SS. FundRatio (FR) >1.1 67/70yo

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Re: Can you make money betting on random coin flips?
Of course
Last edited by TradingPlaces on Fri Jan 23, 2015 11:16 pm, edited 1 time in total.

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Re: Can you make money betting on random coin flips?
If you want to read about the more powerful variant of this,
Last edited by TradingPlaces on Fri Jan 23, 2015 11:16 pm, edited 1 time in total.

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Re: Can you make money betting on random coin flips?
You can only make money if someone is foolish enough to cover the bet.
On the first coin flip you have a 50% chance of making $100 and a 50% chance of losing $50. I'm not covering that bet from the other side. Every subsequent flip has the same odds and proportion of winnings/losses.
On the first coin flip you have a 50% chance of making $100 and a 50% chance of losing $50. I'm not covering that bet from the other side. Every subsequent flip has the same odds and proportion of winnings/losses.

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Re: Can you make money betting on random coin flips?
I'm really sorry if this is ignorant of me, but what sort of investment would give you a double profit:loss for what are essentially 1:1 odds in the long run? Wouldn't best case scenario be 1:1 ProfitLoss? Example if you stand to win $100, you also stand to lose $100, not $50. My understanding is in real life using your same flips the results would be:
Beginning Stake = $100
Flip 1: Win = $200
Flip 2: Lose = $100
Flip 3: Lose = $0
Flip 4: Nothing to wager.
Starting Value = $100
Flip 1: Win = $150 (double your $50 stake + the $50 not wagered)
Flip 2: Lose = $100
Flip 3: Lose = $50
Flip 4: Win = $100
I'm sorry if I'm way off track, I'm only an amateur boglehead at this stage
Beginning Stake = $100
Flip 1: Win = $200
Flip 2: Lose = $100
Flip 3: Lose = $0
Flip 4: Nothing to wager.
Starting Value = $100
Flip 1: Win = $150 (double your $50 stake + the $50 not wagered)
Flip 2: Lose = $100
Flip 3: Lose = $50
Flip 4: Win = $100
I'm sorry if I'm way off track, I'm only an amateur boglehead at this stage
Re: Can you make money betting on random coin flips?
If you flip those coins forever, there is a probability of 1 that at some time in the future you will be up (or down) by any finite number.
If you have significantly more money than the person you are betting against, it is likely that they will run out of money before you do. That is how some casinos make their money... they can play forever because they have so much more money than you.
When gambling, you must be certain to set a limit to your winnings. No matter how much you may be ahead, if you play long enough you will be behind at some point.
vtMaps
If you have significantly more money than the person you are betting against, it is likely that they will run out of money before you do. That is how some casinos make their money... they can play forever because they have so much more money than you.
When gambling, you must be certain to set a limit to your winnings. No matter how much you may be ahead, if you play long enough you will be behind at some point.
vtMaps
"Truly, whoever can make you believe absurdities can make you commit atrocities" Voltaire, as translated by Norman Lewis Torrey
Re: Can you make money betting on random coin flips?
^ ...which is a flaw in your logic Browser. The first game (double all or halve all) might be considered by many to be a very attractive gamble as well. The outcome of that game is equal to $100*(2^n) where n is the difference between the number of heads (you win) and tails (you lose) tossed during the game, irrespective of sequence, as you stated in your op. If the number of heads equals the number of tails, then n=0 and you get your $100 back. Not very interesting, again, as you stated. But the difference n does not converge to zero (equal number of h's and t's) with a large number of tosses as you imply in your op. On the contrary, the longer you play the game, the difference n "randomly walks" to ever larger negative and positive values. So the upshot is that the outcome distribution for such a game played over a large number of tosses is highly bimodal and asymmetric. When n is large and negative (more tails than heads) then you lose (nearly) all of your starting amount ($100). But when n is large and positive, then you win, and sky is the limit for a large number of tosses. As the number of tosses approaches infinity, you have a 50% chance of losing $100 and a 50% chance of winning infinity $'s. I'd say that's a pretty good, but rather time consuming gamble!mindbogle wrote:....and BTW, the number of heads equaling the number of tails does not become "more probable as the number of coin tosses increases". The ratio of heads to tails converges to 1 as the number of tosses increases, but the difference in the number of heads to tails diverges to infinity as the number of coin tosses approaches infinity  one reason, aside from others, that you shouldn't hang out at the casino until you win you money back!Browser wrote: If you were to invest a "lump sum" at the beginning and then just allow it to compound with each coin toss you'd end up with the sum of money you began with, no matter what the sequence of coin tosses, if there are an equal number of heads and tails (which becomes more probable as the number of coin tosses increases). Your net investment return would be zero, so it's not much of an investment opportunity.
MB
As darrellr posted, you would have a hard time finding anyone who would take the other side of either of the two games. Fair games would be double or nothing, not double or halve.
MB
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Re: Can you make money betting on random coin flips?
It is very very easy to be deluded by the first example. You can infer that it's a fair game and that you win or lose nothing, in the long run, if you do bet your entire stake every time.
That's wrong. There's no fundamental difference between the two situations. The bettor has an unfair advantage and cannot lose it by, let's say, "failing to rebalance."
OF COURSE you should take a bet that gives you a 50% chance of winning, and you win twice as much as if you than if you lose. The mental problem is in the first example that seems to give the impression that it is a fair game. It's just like Abbott and Costello "proving" that 13 x 7 = 28, or like the various ways of "proving" you have eleven fingersit can get you going.*
Your examples are not a demonstration of a case where an investing strategykeeping half in cash, in this casecreates a positive return from a situation that otherwise does not have it.
The game favors the bettor, period. You can't change that. Your system of bets can affect whether you take more risk and get a higher, but less certain expected reward, or a lower, but more certain reward. But you do not lose your unfair advantage by using the first system. Your expectation is positive, and all you can do is shape the distribution of outcomes.
In fact, just as you'd expect, your expected return is higher if you bet it all every time. The more money you are betting, the more you profit from your unfair advantage.
The problem is very simple: in your first example, you've shown the outcome for equal numbers of wins and losses. You made the mistaken assumption that since the long term average expectation is 50% heads, therefore the longterm average outcome of playing the game will be the same as the outcome of getting 50% heads. This isn't so. It's a highly skewed distribution. On a sequence of 4 flips, or any other, the median outcome, and also the modethe thing that happens most often, is that get an equal number of wins and loss and break even. BUT it is a very skewed distribution, and when you get more
To understand what's happening, tabulate all 16 possibilities for all 16 sequences of heads and tails, not just one with equal numbers of heads and tails, and calculate the final outcome:
WWWW $1600.00
WWWL $400.00
WWLW $400.00
WLWW $400.00
LWWW $400.00
WWLL $100.00
WLWL $100.00
WLLW $100.00
LWWL $100.00
LWLW $100.00
LLWW $100.00
WLLL $25.00
LWLL $25.00
LLWL $25.00
LLLW $25.00
LLLL $6.25
average $244.14, NOT $100.00.
What's happened here, and this is very significant, is that by using an example with equal numbers of heads and tails and not bothering to calculate any others, you've smuggled in an assumption of mean reversionin this case, perfect mean reversion. If we assume perfect mean reversionif we assume that the only possibility is equal numbers of heads and tailsthen, yes, "rebalancing" beats betting the whole stake every time.
If we only wager half our stake every time, then:
WWWW $506.25
WWWL $253.12
WWLW $253.12
WLWW $253.12
LWWW $253.12
WWLL $126.56
WLWL $126.56
WLLW $126.56
LWWL $126.56
LWLW $126.56
LLWW $126.56
WLLL $63.28
LWLL $63.28
LLWL $63.28
LLLW $63.28
LLLL $31.64
average $160.18
Less risk, as shown by the lower range of outcomes, $31.64$506.26 instead of $6.25$1,600. And less expected reward.
Given the choice, I'd do it the second way because I'm riskaverse, but... expressing it in investment terms... keeping half in cash and rebalancing does not create any rebalancing bonus, it is just choosing to adopt a strategy that reduces both risk and return.
*My favorite: first, count the fingers on the left hand and say "10, 9, 8, 7, 6," hold up second hand and say "...and 5 make eleven." Most people have seen this before and it's so obvious that they see through it. So just as they start to say "hey, wait a minute," I then say "OK, I'll prove it by doing it another way. On the left hand, I say "One, two, let's hold back these three" grabbing three fingers momentarily with right hand then releasing, then raising fingers of right hand and counting "4, 5, 6, 7, 8, now we'll count the ones we held back, 9, 10, 11." I've seen adult scratch their heads for a minute over that one.
That's wrong. There's no fundamental difference between the two situations. The bettor has an unfair advantage and cannot lose it by, let's say, "failing to rebalance."
OF COURSE you should take a bet that gives you a 50% chance of winning, and you win twice as much as if you than if you lose. The mental problem is in the first example that seems to give the impression that it is a fair game. It's just like Abbott and Costello "proving" that 13 x 7 = 28, or like the various ways of "proving" you have eleven fingersit can get you going.*
Your examples are not a demonstration of a case where an investing strategykeeping half in cash, in this casecreates a positive return from a situation that otherwise does not have it.
The game favors the bettor, period. You can't change that. Your system of bets can affect whether you take more risk and get a higher, but less certain expected reward, or a lower, but more certain reward. But you do not lose your unfair advantage by using the first system. Your expectation is positive, and all you can do is shape the distribution of outcomes.
In fact, just as you'd expect, your expected return is higher if you bet it all every time. The more money you are betting, the more you profit from your unfair advantage.
The problem is very simple: in your first example, you've shown the outcome for equal numbers of wins and losses. You made the mistaken assumption that since the long term average expectation is 50% heads, therefore the longterm average outcome of playing the game will be the same as the outcome of getting 50% heads. This isn't so. It's a highly skewed distribution. On a sequence of 4 flips, or any other, the median outcome, and also the modethe thing that happens most often, is that get an equal number of wins and loss and break even. BUT it is a very skewed distribution, and when you get more
To understand what's happening, tabulate all 16 possibilities for all 16 sequences of heads and tails, not just one with equal numbers of heads and tails, and calculate the final outcome:
WWWW $1600.00
WWWL $400.00
WWLW $400.00
WLWW $400.00
LWWW $400.00
WWLL $100.00
WLWL $100.00
WLLW $100.00
LWWL $100.00
LWLW $100.00
LLWW $100.00
WLLL $25.00
LWLL $25.00
LLWL $25.00
LLLW $25.00
LLLL $6.25
average $244.14, NOT $100.00.
What's happened here, and this is very significant, is that by using an example with equal numbers of heads and tails and not bothering to calculate any others, you've smuggled in an assumption of mean reversionin this case, perfect mean reversion. If we assume perfect mean reversionif we assume that the only possibility is equal numbers of heads and tailsthen, yes, "rebalancing" beats betting the whole stake every time.
If we only wager half our stake every time, then:
WWWW $506.25
WWWL $253.12
WWLW $253.12
WLWW $253.12
LWWW $253.12
WWLL $126.56
WLWL $126.56
WLLW $126.56
LWWL $126.56
LWLW $126.56
LLWW $126.56
WLLL $63.28
LWLL $63.28
LLWL $63.28
LLLW $63.28
LLLL $31.64
average $160.18
Less risk, as shown by the lower range of outcomes, $31.64$506.26 instead of $6.25$1,600. And less expected reward.
Given the choice, I'd do it the second way because I'm riskaverse, but... expressing it in investment terms... keeping half in cash and rebalancing does not create any rebalancing bonus, it is just choosing to adopt a strategy that reduces both risk and return.
*My favorite: first, count the fingers on the left hand and say "10, 9, 8, 7, 6," hold up second hand and say "...and 5 make eleven." Most people have seen this before and it's so obvious that they see through it. So just as they start to say "hey, wait a minute," I then say "OK, I'll prove it by doing it another way. On the left hand, I say "One, two, let's hold back these three" grabbing three fingers momentarily with right hand then releasing, then raising fingers of right hand and counting "4, 5, 6, 7, 8, now we'll count the ones we held back, 9, 10, 11." I've seen adult scratch their heads for a minute over that one.
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Re: Can you make money betting on random coin flips?
It appears that the connection with Claude Shannon may be explained in a book which I haven't read and probably should: Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, by William J. Poundstone, 2006.
In 1956 two Bell Labs scientists discovered the scientific formula for getting rich. One was mathematician Claude Shannon, neurotic father of our digital age, whose genius is ranked with Einstein's. The other was John L. Kelly Jr., a Texasborn, guntoting physicist. Together they applied the science of information theorythe basis of computers and the Internetto the problem of making as much money as possible, as fast as possible.
Shannon and MIT mathematician Edward O. Thorp took the "Kelly formula" to Las Vegas. It worked. They realized that there was even more money to be made in the stock market. Thorp used the Kelly system with his phenomenonally successful hedge fund, PrincetonNewport Partners. Shannon became a successful investor, too, topping even Warren Buffett's rate of return. Fortune's Formula traces how the Kelly formula sparked controversy even as it made fortunes at racetracks, casinos, and trading desks. It reveals the dark side of this alluring scheme, which is founded on exploiting an insider's edge.
Shannon believed it was possible for a smart investor to beat the marketand Fortune's Formula will convince you that he was right.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: Can you make money betting on random coin flips?
Correction made.market timer wrote:It's a zero expected geometric return. The arithmetic expected return is .5(100%) + .5(50%) = 25%.Browser wrote:It shows it is possible to make money from an "asset" with random outcomes (coin tossing) that has a zero arithmetic expected return.
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Re: Can you make money betting on random coin flips?
And just to debunk the common fallacy that the number of heads approaches the number of tails after a large number of fair coin tosses.... The graphs in the top row (orange bars) below show histograms of the difference, n, between the resulting number of heads and number of tails after 10, 50, and 200 consecutive toss experiments, each repeated 10,000 times. Note that the larger the number of tosses, the greater the dispersion of n.
If n converged to zero with a large number of tosses (equal number of heads as tails), then one would be justified in expecting that game 1 (double all, or halve all) is a wash since its outcome to the bettor is $100*(2^n). But that growing dispersion in n with number of tosses serves to magnify, not diminish, the asymmetric advantage the bettor has with only risking half of her bet for double return. The bottom row (blue bars) show the distribution of outcomes to the bettor in $'s along the horizontal axis (logarithmic scale). Note that as the number of tosses is increased, the outcome distribution becomes much more dispersed and skewed (even in logarithmic scale!).
Its just math (and a little statistics).
MB
If n converged to zero with a large number of tosses (equal number of heads as tails), then one would be justified in expecting that game 1 (double all, or halve all) is a wash since its outcome to the bettor is $100*(2^n). But that growing dispersion in n with number of tosses serves to magnify, not diminish, the asymmetric advantage the bettor has with only risking half of her bet for double return. The bottom row (blue bars) show the distribution of outcomes to the bettor in $'s along the horizontal axis (logarithmic scale). Note that as the number of tosses is increased, the outcome distribution becomes much more dispersed and skewed (even in logarithmic scale!).
Its just math (and a little statistics).
MB
Re: Can you make money betting on random coin flips?
You're correct  both are unrealistic games with big advantage to the bettor. In my view, the logic error was thinking (or portraying) that the first game was fair because the number of heads approaches the number of tails after many tosses. That's not true.johnnyabardi wrote:I'm really sorry if this is ignorant of me, but what sort of investment would give you a double profit:loss for what are essentially 1:1 odds in the long run? Wouldn't best case scenario be 1:1 ProfitLoss?
MB
Re: Can you make money betting on random coin flips?
The "game" is not contrived. Plenty of investments have a positive expectation, but have risk (:=variance), so the geometric mean is lower than the arithmetic mean. The trick is to diversify so as to try to maximise expected geometric mean of returns, since that is what matters for long term growth.
In the OP's situation you have two investments "cash" (keeps its value) and "bet" (an investment(wager) of amount A has 50% chance of becoming 0.5*A and 50% chance of becoming 2*A). At each time step you want to choose the optimal value of x so that you "invest" x*yourbankroll in "bet", and (1x)*yourbankroll in "cash". "Optimal" means you want to maximze the geometric mean of returns
sqrt[(10.5x)(1+x)]
and basic calculus give you that the maximum is when x=0.5. In other words, always bet half your bankroll in this specific scenario, which is also what the Kelly criterion tells you. In other words your optimal "asset allocation" is 50% "cash" and 50% "bet" and you "rebalance" at each step.
It amounts to a very simplified example of why we diversify, except in the real world we don't know the probability distribution of returns.
In the OP's situation you have two investments "cash" (keeps its value) and "bet" (an investment(wager) of amount A has 50% chance of becoming 0.5*A and 50% chance of becoming 2*A). At each time step you want to choose the optimal value of x so that you "invest" x*yourbankroll in "bet", and (1x)*yourbankroll in "cash". "Optimal" means you want to maximze the geometric mean of returns
sqrt[(10.5x)(1+x)]
and basic calculus give you that the maximum is when x=0.5. In other words, always bet half your bankroll in this specific scenario, which is also what the Kelly criterion tells you. In other words your optimal "asset allocation" is 50% "cash" and 50% "bet" and you "rebalance" at each step.
It amounts to a very simplified example of why we diversify, except in the real world we don't know the probability distribution of returns.
Re: Can you make money betting on random coin flips?
I have to admit I have no idea what some posters are talking about. With a sufficiently large number of coin flips, the expectation is for an equal number of heads and tails. To my knowledge the law of large numbers has not been repealed. Therefore, the cointoss game has zero longterm expected growth. Shannon's Demon, though it is a contrived situation, is a demonstration that it is possible to generate positive growth with a multiasset "portfolio" which includes the cointoss game, which is an "asset" having no expected longterm growth. This is true to the extent that total "portfolio" returns are not dominated by the absolute returns of the zeroreturn asset, which becomes less and less likely as the series of coin tosses gets longer. By analogy it demonstrates the phenomenon of "volatility harvesting" and the benefits of asset allocation rebalancing, particularly when no single asset dominates portfolio returns.
We don't know where we are, or where we're going  but we're making good time.
Re: Can you make money betting on random coin flips?
Yes, understood. But that's not how the OP presented the problem, the way I read it. It wasn't about maximizing returns as it was about making money off of what "appeared" to be zero expected value game, as what was implied for the first game. I don't think I am the only one to read it that way.555 wrote:The "game" is not contrived. Plenty of investments have a positive expectation, but have risk (:=variance), so the geometric mean is lower than the arithmetic mean. The trick is to diversify so as to try to maximise expected geometric mean of returns, since that is what matters for long term growth.
In the OP's situation you have two investments "cash" (keeps its value) and "bet" (an investment(wager) of amount A has 50% chance of becoming 0.5*A and 50% chance of becoming 2*A). At each time step you want to choose the optimal value of x so that you "invest" x*yourbankroll in "bet", and (1x)*yourbankroll in "cash". "Optimal" means you want to maximze the geometric mean of returns
sqrt[(10.5x)(1+x)]
and basic calculus give you that the maximum is when x=0.5. In other words, always bet half your bankroll in this specific scenario, which is also what the Kelly criterion tells you. In other words your optimal "asset allocation" is 50% "cash" and 50% "bet" and you "rebalance" at each step.
It amounts to a very simplified example of why we diversify, except in the real world we don't know the probability distribution of returns.
MB
Re: Can you make money betting on random coin flips?
No, with sufficiently large number of coin flips, the expectation is that the ratio of heads to tails will converge to 1, or that the percent of heads will converge to 50%, not that the number of heads will get closer to the number of tails. The "nuanced" difference is important, especially to the way you presented the original game problem. Its a common fallacy. I presented simulation results showing what happens to the difference between number of heads to tails as the number of coin flips gets larger  it grows, doesn't shrink. That difference is what drives your game outcomes. But its easy enough to try for yourself and prove me wrong!Browser wrote:I have to admit I have no idea what some posters are talking about. With a sufficiently large number of coin flips, the expectation is for an equal number of heads and tails. To my knowledge the law of large numbers has not been repealed. Therefore, the cointoss game has zero longterm expected growth. Shannon's Demon, though it is a contrived situation, is a demonstration that it is possible to generate positive growth with a multiasset "portfolio" which includes the cointoss game, which is an "asset" having no expected longterm growth. This is true to the extent that total "portfolio" returns are not dominated by the absolute returns of the zeroreturn asset, which becomes less and less likely as the series of coin tosses gets longer. By analogy it demonstrates the phenomenon of "volatility harvesting" and the benefits of asset allocation rebalancing, particularly when no single asset dominates portfolio returns.
MB
Re: Can you make money betting on random coin flips?
The game has no expected gain if you use geometric mean. sqrt[0.5*2]=1
The game has an expected gain if you use arithmetic mean. [0.5+2]/2=1.25
This is all very old news. It's an interesting topic, but the OP does not seem to understand or communicate it well.
The game has an expected gain if you use arithmetic mean. [0.5+2]/2=1.25
This is all very old news. It's an interesting topic, but the OP does not seem to understand or communicate it well.
Re: Can you make money betting on random coin flips?
The OP has misunderstood the law of large numbers, and appeared to be assuming a fallacious version of "reversion to mean", but nevertheless the overall conclusion is essentially right.mindbogle wrote:No, with sufficiently large number of coin flips, the expectation is that the ratio of heads to tails will converge to 1, or that the percent of heads will converge to 50%, not that the number of heads will get closer to the number of tails. The "nuanced" difference is important, especially to the way you presented the original game problem. Its a common fallacy. I presented simulation results showing what happens to the difference between number of heads to tails as the number of coin flips gets larger  it grows, doesn't shrink. That difference is what drives your game outcomes. But its easy enough to try for yourself and prove me wrong!Browser wrote:I have to admit I have no idea what some posters are talking about. With a sufficiently large number of coin flips, the expectation is for an equal number of heads and tails. To my knowledge the law of large numbers has not been repealed. Therefore, the cointoss game has zero longterm expected growth. Shannon's Demon, though it is a contrived situation, is a demonstration that it is possible to generate positive growth with a multiasset "portfolio" which includes the cointoss game, which is an "asset" having no expected longterm growth. This is true to the extent that total "portfolio" returns are not dominated by the absolute returns of the zeroreturn asset, which becomes less and less likely as the series of coin tosses gets longer. By analogy it demonstrates the phenomenon of "volatility harvesting" and the benefits of asset allocation rebalancing, particularly when no single asset dominates portfolio returns.
MB
Re: Can you make money betting on random coin flips?
I think nisiprius illustrated it very well. The median result of all the possible outcomes for number of coin flips may be 0 return, but the average return is much higher because if you're fortunate enough to win more bets than you lose, your winnings are substantially higher than your losses would be in the case that you lost more coin flips. Look at it this way, if you made this bet 1 million times and won 500,000 times and lost 500,000 times, you'd break even. But what if your actual results were 500,005 wins and 499,995 losses? You would've doubled your stake an extra 10 times, meaning you'd have 1024 times your initial amount! Of course, you're equally likely to see the reverse, 499,995 wins and 500,005 losses, which would be an extra 10 "halvings", leaving you with only 1/1024 your initial stake, but the average of those 2 outcomes is still more than 500 times your initial stake. I don't feel like digging out my normal distribution tables, but deviating from the expected 50/50 split by 5 only samples in a million, one way or the other, is extraordinarily likely. What you're doing with the strategy of setting aside some of your balance each time, instead of continually betting it all, is reducing your variance so that instead of having a 50/50 chance of overperforming expectation by a lot or underperforming expectation by a a little, you accept a more reliable return of close to 25% of the wagered amount per flip, which is exactly what you'd expect from a game in which you gain x for a win and only lose 0.5x for loss.Browser wrote:I have to admit I have no idea what some posters are talking about. With a sufficiently large number of coin flips, the expectation is for an equal number of heads and tails. To my knowledge the law of large numbers has not been repealed. Therefore, the cointoss game has zero longterm expected growth. Shannon's Demon, though it is a contrived situation, is a demonstration that it is possible to generate positive growth with a multiasset "portfolio" which includes the cointoss game, which is an "asset" having no expected longterm growth. This is true to the extent that total "portfolio" returns are not dominated by the absolute returns of the zeroreturn asset, which becomes less and less likely as the series of coin tosses gets longer. By analogy it demonstrates the phenomenon of "volatility harvesting" and the benefits of asset allocation rebalancing, particularly when no single asset dominates portfolio returns.
Last edited by Ungoliant on Mon Jan 05, 2015 2:06 pm, edited 3 times in total.
Re: Can you make money betting on random coin flips?
Just for reference:
I fail to see how I have "misunderstood" this.
http://en.wikipedia.org/wiki/Law_of_large_numbersIn probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
I fail to see how I have "misunderstood" this.
We don't know where we are, or where we're going  but we're making good time.
Re: Can you make money betting on random coin flips?
Perhaps you could explain the scenario by explicitly spelling out the probability distribution of outcomes after n steps.Browser wrote:Just for reference:http://en.wikipedia.org/wiki/Law_of_large_numbersIn probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
I fail to see how I have "misunderstood" this.
You could clearly explain to readers the difference between different kinds of means, and how that applies to this situation.
You could compare and contrast fact and fiction when it comes to phrases like "reversion to the mean".
Last edited by 555 on Mon Jan 05, 2015 2:31 pm, edited 1 time in total.
Re: Can you make money betting on random coin flips?
This deal is actually not an even bet at all. If you're given the option of taking this bet, you should ABSOLUTELY take the bet, and you should bet 100% on it. More accurately, you should split your money into 100 seperate bets and place all of them. Laws of probability say you'll get an almost guaranteed 25% return. Look at it this way:
Start with $100
Make 100 bets of $1
50 bets you are correct, and you get $2 each per bet for a total of $100 for those.
50 bets you are incorrect, and you get $0.50 each per bet for a total of $25.
Total winnings: $125.
Do that again and again until the counterparty is broke or breaks your nose.
This is a bet that is already in your favor.
As for the 26% return the OP showed, it's actually simply a reworking of the actual return it provides, which is 12.5%. It isn't giving you a free 1%. Every win gets you 50% return on your total balance, every loss gets you a 25% loss on your total balance. So 1.5 x 0.75 = 1.125. Therefore you are expected to make a 12.5% return on average per bet. Which is actually perfectly reasonable. Why? because you're only betting half your money, your 12.5% return is exactly half of what you'd be getting if you bet everything, 25%. Because the OP made two wins and two losses, this had the net effect of compounding that 12.5% return. So 1.125 x 1.125 = 1.265625. So it looks pretty impressive.
But what OP actually did was take a 25 percent return, cut it in half, then compound it twice. By just betting everything, your expected return stays at 25%, much better than the 12.5% that OP wanted to give you. And if you do it twice, as OP did, you get 56.25% not a lousy 26%.
Start with $100
Make 100 bets of $1
50 bets you are correct, and you get $2 each per bet for a total of $100 for those.
50 bets you are incorrect, and you get $0.50 each per bet for a total of $25.
Total winnings: $125.
Do that again and again until the counterparty is broke or breaks your nose.
This is a bet that is already in your favor.
As for the 26% return the OP showed, it's actually simply a reworking of the actual return it provides, which is 12.5%. It isn't giving you a free 1%. Every win gets you 50% return on your total balance, every loss gets you a 25% loss on your total balance. So 1.5 x 0.75 = 1.125. Therefore you are expected to make a 12.5% return on average per bet. Which is actually perfectly reasonable. Why? because you're only betting half your money, your 12.5% return is exactly half of what you'd be getting if you bet everything, 25%. Because the OP made two wins and two losses, this had the net effect of compounding that 12.5% return. So 1.125 x 1.125 = 1.265625. So it looks pretty impressive.
But what OP actually did was take a 25 percent return, cut it in half, then compound it twice. By just betting everything, your expected return stays at 25%, much better than the 12.5% that OP wanted to give you. And if you do it twice, as OP did, you get 56.25% not a lousy 26%.
Re: Can you make money betting on random coin flips?
Because it's a growth problem, the correct solution is lognormal. Here's the equation:
Growth Rate = 0.5*LN(1+1) + 0.5*LN(10.5) = 0
For the case where just half of the current stake is wagered on each coin toss, the solution is:
Growth Rate = 0.5*LN(1+0.5) + 0.5*LN(10.25) = 0.059
Growth Rate = 0.5*LN(1+1) + 0.5*LN(10.5) = 0
For the case where just half of the current stake is wagered on each coin toss, the solution is:
Growth Rate = 0.5*LN(1+0.5) + 0.5*LN(10.25) = 0.059
Last edited by Browser on Mon Jan 05, 2015 2:38 pm, edited 2 times in total.
We don't know where we are, or where we're going  but we're making good time.
Re: Can you make money betting on random coin flips?
Browser, your problem with law of large numbers is that it deals with the expectation of a sum, or arithmetic mean. You can't use it if you are not adding, but multiplying random variables like you do for a geometric mean.
Re: Can you make money betting on random coin flips?
Math does not check out heretoto238 wrote:This deal is actually not an even bet at all. If you're given the option of taking this bet, you should ABSOLUTELY take the bet, and you should bet 100% on it. More accurately, you should split your money into 100 seperate bets and place all of them. Laws of probability say you'll get an almost guaranteed 25% return. Look at it this way:
Start with $100
Make 100 bets of $1
50 bets you are correct, and you get $2 each per bet for a total of $100 for those.
50 bets you are incorrect, and you get $0.50 each per bet for a total of $25.
Total winnings: $125.
Do that again and again until the counterparty is broke or breaks your nose.
This is a bet that is already in your favor.
As for the 26% return the OP showed, it's actually simply a reworking of the actual return it provides, which is 12.5%. It isn't giving you a free 1%. Every win gets you 50% return on your total balance, every loss gets you a 25% loss on your total balance. So 1.5 x 0.75 = 1.125. Therefore you are expected to make a 12.5% return on average per bet. Which is actually perfectly reasonable. Why? because you're only betting half your money, your 12.5% return is exactly half of what you'd be getting if you bet everything, 25%. Because the OP made two wins and two losses, this had the net effect of compounding that 12.5% return. So 1.125 x 1.125 = 1.265625. So it looks pretty impressive.
But what OP actually did was take a 25 percent return, cut it in half, then compound it twice. By just betting everything, your expected return stays at 25%, much better than the 12.5% that OP wanted to give you. And if you do it twice, as OP did, you get 56.25% not a lousy 26%.
Re: Can you make money betting on random coin flips?
No that's not a problem.aaasdaef wrote:Browser, your problem with law of large numbers is that it deals with the expectation of a sum, or arithmetic mean. You can't use it if you are not adding, but multiplying random variables like you do for a geometric mean.
Re: Can you make money betting on random coin flips?
I would not call it an explicit misunderstanding, but here's the difference that 555 is talking about:Browser wrote:I fail to see how I have "misunderstood" this.
Let D be the expected absolute difference between heads and tails. After N trials:
Law of large numbers: D/N converges to zero.
Incorrect reading (may or may not be yours): D converges to zero. In reality, D grows with N but not as fast as N.
For any absolute D, negative and positive differences are equally likely. The payouts are not: your profit is S x 2^(D  1), D your loss is S / 2^(D1), which is at most the starting sum S. Therefore, your gain expectation increases with D, so it increases with N and you should also seek to maximize the amount invested.
The right way to play the game is actually to go allin with every flip. This does not virtually guarantee a profit like your rebalancelike strategy, but it maximizes the profit expectation because when the profits do happen they are much higher than in the safer strategy. Higher than the customary linear risk/return, I think, though I haven't fully done the math.
Last edited by ogd on Mon Jan 05, 2015 2:43 pm, edited 1 time in total.
Re: Can you make money betting on random coin flips?
Please tell me where my math has erred. I will gladly correct it. As it is, you have simply made an accusation without backing it up with anything at all.fanmail wrote:Math does not check out heretoto238 wrote:This deal is actually not an even bet at all. If you're given the option of taking this bet, you should ABSOLUTELY take the bet, and you should bet 100% on it. More accurately, you should split your money into 100 seperate bets and place all of them. Laws of probability say you'll get an almost guaranteed 25% return. Look at it this way:
Start with $100
Make 100 bets of $1
50 bets you are correct, and you get $2 each per bet for a total of $100 for those.
50 bets you are incorrect, and you get $0.50 each per bet for a total of $25.
Total winnings: $125.
Do that again and again until the counterparty is broke or breaks your nose.
This is a bet that is already in your favor.
As for the 26% return the OP showed, it's actually simply a reworking of the actual return it provides, which is 12.5%. It isn't giving you a free 1%. Every win gets you 50% return on your total balance, every loss gets you a 25% loss on your total balance. So 1.5 x 0.75 = 1.125. Therefore you are expected to make a 12.5% return on average per bet. Which is actually perfectly reasonable. Why? because you're only betting half your money, your 12.5% return is exactly half of what you'd be getting if you bet everything, 25%. Because the OP made two wins and two losses, this had the net effect of compounding that 12.5% return. So 1.125 x 1.125 = 1.265625. So it looks pretty impressive.
But what OP actually did was take a 25 percent return, cut it in half, then compound it twice. By just betting everything, your expected return stays at 25%, much better than the 12.5% that OP wanted to give you. And if you do it twice, as OP did, you get 56.25% not a lousy 26%.
Re: Can you make money betting on random coin flips?
I think I'll retitle this thread to "Fun With Numbers"
We don't know where we are, or where we're going  but we're making good time.
Re: Can you make money betting on random coin flips?
If you bet $1, you either end up with $2 if you win or $0 if you lose. Why would you get paid when you lose lol?toto238 wrote:Please tell me where my math has erred. I will gladly correct it. As it is, you have simply made an accusation without backing it up with anything at all.fanmail wrote:Math does not check out heretoto238 wrote:This deal is actually not an even bet at all. If you're given the option of taking this bet, you should ABSOLUTELY take the bet, and you should bet 100% on it. More accurately, you should split your money into 100 seperate bets and place all of them. Laws of probability say you'll get an almost guaranteed 25% return. Look at it this way:
Start with $100
Make 100 bets of $1
50 bets you are correct, and you get $2 each per bet for a total of $100 for those.
50 bets you are incorrect, and you get $0.50 each per bet for a total of $25.
Total winnings: $125.
Do that again and again until the counterparty is broke or breaks your nose.
This is a bet that is already in your favor.
As for the 26% return the OP showed, it's actually simply a reworking of the actual return it provides, which is 12.5%. It isn't giving you a free 1%. Every win gets you 50% return on your total balance, every loss gets you a 25% loss on your total balance. So 1.5 x 0.75 = 1.125. Therefore you are expected to make a 12.5% return on average per bet. Which is actually perfectly reasonable. Why? because you're only betting half your money, your 12.5% return is exactly half of what you'd be getting if you bet everything, 25%. Because the OP made two wins and two losses, this had the net effect of compounding that 12.5% return. So 1.125 x 1.125 = 1.265625. So it looks pretty impressive.
But what OP actually did was take a 25 percent return, cut it in half, then compound it twice. By just betting everything, your expected return stays at 25%, much better than the 12.5% that OP wanted to give you. And if you do it twice, as OP did, you get 56.25% not a lousy 26%.
Re: Can you make money betting on random coin flips?
It's fun to think about the "triple or nothing" coin flip bet, which has been discussed in past threads (bet $1, get $3(or your $1 back plus $2 profit) if you win and 0$(lose your $1 bet) if you lose).
If you bet your whole bankroll and let it ride, you'll almost definitely go broke. The optimum is to bet 25% or your bankroll.
If you bet your whole bankroll and let it ride, you'll almost definitely go broke. The optimum is to bet 25% or your bankroll.
 Epsilon Delta
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 Joined: Thu Apr 28, 2011 7:00 pm
Re: Can you make money betting on random coin flips?
What you have misunderstood is that "The average converges to the expected value" is not the same as "The total converges to n times the expected value".Browser wrote:Just for reference:http://en.wikipedia.org/wiki/Law_of_large_numbersIn probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
I fail to see how I have "misunderstood" this.

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Re: Can you make money betting on random coin flips?
Had to run home to get my Tablet and check the Market.
It's a Great Day
As for this topic;
The Rich can get richer.
A short run can be very profitable.
The Market is not a set of dice or a HeadsTails.
The Market can be very good if you can see some future trends.
The longer you play in the Market, you best be in The Index.
I reentered the Market on Friday, a bit too much and a bit too soon. Exited some holdings today with a fair loss but tolerable and should have exited earlier this AM when it looked a little bad.
It's a Great Day
As for this topic;
The Rich can get richer.
A short run can be very profitable.
The Market is not a set of dice or a HeadsTails.
The Market can be very good if you can see some future trends.
The longer you play in the Market, you best be in The Index.
I reentered the Market on Friday, a bit too much and a bit too soon. Exited some holdings today with a fair loss but tolerable and should have exited earlier this AM when it looked a little bad.
Rev012718; 4 Incm stream buckets: SS+pension; dfr'd GLWB VA & FI anntys, by time & $$ laddered; Discretionary; Rentals. LTCi. Own, not asset. Tax TBT%. Early SS. FundRatio (FR) >1.1 67/70yo
Re: Can you make money betting on random coin flips?
The example given suggests a scenario where a winning bet pays odds 2:1 for a coin flip, where fair odds would only pay out 1:1.
This is silly because nobody in their right mind would offer that payout for very long (they would go broke). Of course a rational "investor" should use some moneymanagement technique like the Kelly Criterion to maximize the opportunity and it suggests you should wager half the bankroll. But if the odds were paid out fairly 1:1 the Kelly formula would tell you not to wager at all, there would be no advantage.. only risk.. and you shouldn't play the game.
This is silly because nobody in their right mind would offer that payout for very long (they would go broke). Of course a rational "investor" should use some moneymanagement technique like the Kelly Criterion to maximize the opportunity and it suggests you should wager half the bankroll. But if the odds were paid out fairly 1:1 the Kelly formula would tell you not to wager at all, there would be no advantage.. only risk.. and you shouldn't play the game.
"To achieve satisfactory investment results is easier than most people realize; to achieve superior results is harder than it looks."  Benjamin Graham
Re: Can you make money betting on random coin flips?
toto238's calculation is correct (it uses correctly the bet as described in the OP), but it's a different scenario where you can make multiple bets in parallel, not just in series as in the OP.toto238 wrote:Please tell me where my math has erred. I will gladly correct it. As it is, you have simply made an accusation without backing it up with anything at all.fanmail wrote:Math does not check out heretoto238 wrote:This deal is actually not an even bet at all. If you're given the option of taking this bet, you should ABSOLUTELY take the bet, and you should bet 100% on it. More accurately, you should split your money into 100 seperate bets and place all of them. Laws of probability say you'll get an almost guaranteed 25% return. Look at it this way:
Start with $100
Make 100 bets of $1
50 bets you are correct, and you get $2 each per bet for a total of $100 for those.
50 bets you are incorrect, and you get $0.50 each per bet for a total of $25.
Total winnings: $125.
Do that again and again until the counterparty is broke or breaks your nose.
This is a bet that is already in your favor.
As for the 26% return the OP showed, it's actually simply a reworking of the actual return it provides, which is 12.5%. It isn't giving you a free 1%. Every win gets you 50% return on your total balance, every loss gets you a 25% loss on your total balance. So 1.5 x 0.75 = 1.125. Therefore you are expected to make a 12.5% return on average per bet. Which is actually perfectly reasonable. Why? because you're only betting half your money, your 12.5% return is exactly half of what you'd be getting if you bet everything, 25%. Because the OP made two wins and two losses, this had the net effect of compounding that 12.5% return. So 1.125 x 1.125 = 1.265625. So it looks pretty impressive.
But what OP actually did was take a 25 percent return, cut it in half, then compound it twice. By just betting everything, your expected return stays at 25%, much better than the 12.5% that OP wanted to give you. And if you do it twice, as OP did, you get 56.25% not a lousy 26%.
Re: Can you make money betting on random coin flips?
Ah, my mistake. The ol lose only half when you lose but make double when you win bet. Yes, of course you wold want to do that. Probably use Kelly criterion to find the optimal bet sizing.
 nisiprius
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Re: Can you make money betting on random coin flips?
My thoughts, subject to correction. The arithmetic mean and geometric mean both are appropriate in investing in different situations. The geometric mean is what applies when you are concatenating a series of investment results in time: 2012, 2013, 2014, etc. The appropriate "average" for the annual return of a threeyear investment is the geometric mean of each years' returns, and the reason is that growing by the amount of the geometric mean in 2012, then 2013, then 2014 is the same as growing by the actual three different amounts.555 wrote:No that's not a problem.aaasdaef wrote:Browser, your problem with law of large numbers is that it deals with the expectation of a sum, or arithmetic mean. You can't use it if you are not adding, but multiplying random variables like you do for a geometric mean.
The (weighted) arithmetic mean is appropriate when you are talking about the returns of an mix of investments occurring simultaneously over a single period of time. It can be an actual mix as in a portfolio, or it can be a statistical mix of hypothetical scenarios (as when talking about expected return).
Thus, if we assume that we have pure assets A and B, and that we are talking about a period of time in which we let them run without rebalancinglet's say for one year, and if the actual annual return for the two assets is 4% and 6% respectively$10,000 invested in asset A is worth $10,400 at the end of the year and $10,000 invested in asset B is worth $10,600 at the end of the yearthen the return of a portfolio of 50% A, 50% B is just the plain old arithmetic mean, 5%.
Now, that total annual return of 6% for the whole year might be considered to be the result of compounding happening during the year, but nevertheless it is true and is just plain arithmeticyou combine the 4% and the 6% as a simple arithmetic mean.
It can be a statistical mix, too. If, at the start of the year, we invest $10,000 in either asset A or B but we flip a coin to decide which, then our expected return for the year is just the plain old arithmetic mean of 4% and 6% = 5%.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: Can you make money betting on random coin flips?
Thank you. Yes according to OP, the rules of the bet are that you get 50% of your money back even if you were wrong.fanmail wrote:Ah, my mistake. The ol lose only half when you lose but make double when you win bet. Yes, of course you wold want to do that. Probably use Kelly criterion to find the optimal bet sizing.
Fantastic bet in my opinion.
Re: Can you make money betting on random coin flips?
I guess what I am trying to say is that an expectation is defined as a sum of probability * outcome. So, as people pointed out, your expected value is greater than 1. There's no "geometric" expected value that's done in the log domain. You'd only do that if your target is not dollars, but logdollars. For instance, some economists argue that utility of money is in logdollars. So, if losing 100 times in a row, and having near 0 dollars means you starve, and has utility of "almost negative infinity", that would get to cancel out positive "almost infinity" you get from winning 100 times in a row. This would be a reasonable model if your initial dollar was all you had to live on for the rest of your life. But in this case, it's not.
The expectation of "money" here increases with each toss. The expectation of "log money" is exactly the same after each toss, namely 0. I know it's counterintuitive and that's what's throwing you off.
The expectation of "money" here increases with each toss. The expectation of "log money" is exactly the same after each toss, namely 0. I know it's counterintuitive and that's what's throwing you off.
 nisiprius
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Re: Can you make money betting on random coin flips?
Browser, let me try one more time because this is where you're getting it wrong. Your problem is in your mental model of the first situation, the one where you bet your entire stake every time.
You are correct that the statistically "expected" numbers of heads and tails is equal.
You are correct that getting equal numbers of heads and tails is more probable than any other pair of numbers. For example, in 10 flips, the probability of 5 heads and 5 tails is higher than the probability of 6 heads and 4 tails. In fact the probability of 5 heads and 5 tails is 24.6%; of 6 heads and 4 tails, 20.5%.
You are correct that, when expressed as a percentage, the distribution "tightens up" and becomes narrower as the number of trials increases.
The probability of getting 3 or fewer heads in 10 flips is 17.1% percent. Not unlikely at all.
The probability of getting 6 or fewer heads in 20 flips is 5.7%.
The probability of getting 15 or fewer heads in 50 flips is 0.33%. Rare.
The probability of getting 30 or fewer heads in 100 flips is 0.0039%.
The probability of getting 300 or fewer heads in 1000 flips is 0.00000075%
You are correct that if you have equal numbers of heads and tails in any order, then you do not win or lose any money.
And yet... although you are correct about all these things, you are NOT correct in thinking that your statistical expectation in playing the game, betting your whole stake every time, is zero. The reason is that even though over a long time, the spread in the outcome expressed as "percentage of heads" narrows, it doesn't go awayand the spread in the outcome expressed in dollars is always unbalanced. If you take the case of 20 flips, I think you can agree that 11 wins, 9 losses has exactly the same probability as 9 wins, 11 losses. But:
11 wins, 9 losses > final outcome = $400.
10 wins, 10 losses > final outcome = $100.
9 wins, 11 losses > final outcome $25
And something you might be missing: this relationship remains the sameone single extra win or loss has the same effectno matter how long the number of coin flips and no matter how tight the percentage is.
51 wins, 49 losses > final outcome = $400.
50 wins, 50 losses > final outcome = $100.
49 wins, 51 losses > final outcome $25.
500,001 wins, 499,999 losses > final outcome = $400.
500,000 wins, 500,000 losses > final outcome = $100.
499,999 wins, 500,001 losses > final outcome $25.
The inexorable fact remains: if you play the same with one flip, your expected return is the average of $50 and $200, which is $125. Your expected return is a percentage gain of 25% per flip. No matter how many times you flip, there is always statistically some spread of outcomes, even though it is narrow, you win much more if you get a few extra heads than you lose if you get a few extra tails, and the overall average is still 25% per flip.
If you go back to my fullyworkedout example for four flips, the average outcome is $244.14. Well, 1.25 x 1.25 x 1.25 x 1.25 = 2.4414. The weird thing is that you are not see any of these gains from the most common, most likely case of two heads and two tails. It comes from the imbalance of +$300 with only one extra head, but only $75 for one extra tail, and similarly for +$1500 versus $93.75.
WWWW $1600.00
WWWL $400.00
WWLW $400.00
WLWW $400.00
LWWW $400.00
WWLL $100.00
WLWL $100.00
WLLW $100.00
LWWL $100.00
LWLW $100.00
LLWW $100.00
WLLL $25.00
LWLL $25.00
LLWL $25.00
LLLW $25.00
LLLL $6.25
You are correct that the statistically "expected" numbers of heads and tails is equal.
You are correct that getting equal numbers of heads and tails is more probable than any other pair of numbers. For example, in 10 flips, the probability of 5 heads and 5 tails is higher than the probability of 6 heads and 4 tails. In fact the probability of 5 heads and 5 tails is 24.6%; of 6 heads and 4 tails, 20.5%.
You are correct that, when expressed as a percentage, the distribution "tightens up" and becomes narrower as the number of trials increases.
The probability of getting 3 or fewer heads in 10 flips is 17.1% percent. Not unlikely at all.
The probability of getting 6 or fewer heads in 20 flips is 5.7%.
The probability of getting 15 or fewer heads in 50 flips is 0.33%. Rare.
The probability of getting 30 or fewer heads in 100 flips is 0.0039%.
The probability of getting 300 or fewer heads in 1000 flips is 0.00000075%
You are correct that if you have equal numbers of heads and tails in any order, then you do not win or lose any money.
And yet... although you are correct about all these things, you are NOT correct in thinking that your statistical expectation in playing the game, betting your whole stake every time, is zero. The reason is that even though over a long time, the spread in the outcome expressed as "percentage of heads" narrows, it doesn't go awayand the spread in the outcome expressed in dollars is always unbalanced. If you take the case of 20 flips, I think you can agree that 11 wins, 9 losses has exactly the same probability as 9 wins, 11 losses. But:
11 wins, 9 losses > final outcome = $400.
10 wins, 10 losses > final outcome = $100.
9 wins, 11 losses > final outcome $25
And something you might be missing: this relationship remains the sameone single extra win or loss has the same effectno matter how long the number of coin flips and no matter how tight the percentage is.
51 wins, 49 losses > final outcome = $400.
50 wins, 50 losses > final outcome = $100.
49 wins, 51 losses > final outcome $25.
500,001 wins, 499,999 losses > final outcome = $400.
500,000 wins, 500,000 losses > final outcome = $100.
499,999 wins, 500,001 losses > final outcome $25.
The inexorable fact remains: if you play the same with one flip, your expected return is the average of $50 and $200, which is $125. Your expected return is a percentage gain of 25% per flip. No matter how many times you flip, there is always statistically some spread of outcomes, even though it is narrow, you win much more if you get a few extra heads than you lose if you get a few extra tails, and the overall average is still 25% per flip.
If you go back to my fullyworkedout example for four flips, the average outcome is $244.14. Well, 1.25 x 1.25 x 1.25 x 1.25 = 2.4414. The weird thing is that you are not see any of these gains from the most common, most likely case of two heads and two tails. It comes from the imbalance of +$300 with only one extra head, but only $75 for one extra tail, and similarly for +$1500 versus $93.75.
WWWW $1600.00
WWWL $400.00
WWLW $400.00
WLWW $400.00
LWWW $400.00
WWLL $100.00
WLWL $100.00
WLLW $100.00
LWWL $100.00
LWLW $100.00
LLWW $100.00
WLLL $25.00
LWLL $25.00
LLWL $25.00
LLLW $25.00
LLLL $6.25
Last edited by nisiprius on Mon Jan 05, 2015 3:42 pm, edited 1 time in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: Can you make money betting on random coin flips?
Sequence matters as well if you are betting % vs betting static $ amounts. News at 11.
Re: Can you make money betting on random coin flips?
I'm shocked that a handful of 4toss trials don't precisely fit the theoretically modeled expected values. Are you suggesting that we should just toss out the theoretically modeled values, then? The conclusions drawn from Shannon's "thought experiment" are the important takeaway here.
We don't know where we are, or where we're going  but we're making good time.
Re: Can you make money betting on random coin flips?
With the OP's method, the geometric expected growth is
sqrt[(10.25)(1+0.5)]=sqrt(1.125)
Now sqrt(1.125) is between
1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311795
and
1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311796
So if you start with 1 money unit, then for large enough n, after n steps your balance will be between
(1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311795)^n
and
(1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311796)^n
with probability greater than 0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
sqrt[(10.25)(1+0.5)]=sqrt(1.125)
Now sqrt(1.125) is between
1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311795
and
1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311796
So if you start with 1 money unit, then for large enough n, after n steps your balance will be between
(1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311795)^n
and
(1.0606601717798212866012665431572735589272539065327110548825098034930493588465802791377906507457311796)^n
with probability greater than 0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
Re: Can you make money betting on random coin flips?
I will repeat what others have already expressed in a bit of a different way. Think of this as an expected value problem. For only one coin toss there are two possible outcomes with equal probabilities. If you win you get 200 dollars. If you lose you get 50 dollars. Therefore the expected value of betting 100 dollars on the coin toss is (.5)*200+(.5)*50=125 dollars. The more you wager the more you are likely to gain.I'm shocked that a handful of 4toss trials don't precisely fit the theoretically modeled expected values. Are you suggesting that we should just toss out the theoretically modeled values, then? The conclusions drawn from Shannon's "thought experiment" are the important takeaway here.
I am not sure what conclusions Shannon drew from this thought experiment. The only thing I see this prove is that if you take more risk you get more reward and vice versa. There is nothing new there.