Is this a good or bad bet?
Is this a good or bad bet?
I ran across a discussion of this question in a paper by Paul Samuelson on investment management.
You have the opportunity to bet on repeated flips of a fair coin. For heads, you get a 170% gain (for instance, $100 becomes $270), for tails, you have a 70% loss (for instance, $100 becomes $30).
Would you take the bet?
You have the opportunity to bet on repeated flips of a fair coin. For heads, you get a 170% gain (for instance, $100 becomes $270), for tails, you have a 70% loss (for instance, $100 becomes $30).
Would you take the bet?
 Mister Whale
 Posts: 477
 Joined: Sat Jan 02, 2010 10:39 am
Re: Is this a good or bad bet?
No way!
I'd be curious to see the context of the question and the subsequent discussion.
I'd be curious to see the context of the question and the subsequent discussion.
" ... advice is most useful and at its best, not when it is telling you what to do, but when it is illuminating aspects of the situation you hadn't thought about." nisiprius
Re: Is this a good or bad bet?
Surely, with some nonlifechanging portion of your net worth, you take this bet as many times as possible. Even as you get infinitely close to a total loss, you never get all the way there. I'm obviously missing some higher level insight!

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Re: Is this a good or bad bet?
I'd definitely take the bet, but only with a very small piece of my net worth. The bet has a positive expected value, but you're going to lose more often than you win.
For example, say we flip just two times. There's four possible outcomes: headsheads, headstails, tailsheads, tailstails. If heads is the winning side, we would have the results: $729, $81, $81, $9 AVERAGE: $225
So, good expected value, but don't bet the farm!
For example, say we flip just two times. There's four possible outcomes: headsheads, headstails, tailsheads, tailstails. If heads is the winning side, we would have the results: $729, $81, $81, $9 AVERAGE: $225
So, good expected value, but don't bet the farm!
Re: Is this a good or bad bet?
No. This example is meant to teach you how losing a percent of your money is not equal to gaining that same percentage back. In order to recover from a 70% loss, you need to gain 233.33%. 170% won't cut it; you are guaranteed to lose money.
Another way to look at it is 75% of the time you lose money, 25% of the time you make money. Now, would you still take that bet?
Scandinavian wrote:For example, say we flip just two times. There's four possible outcomes: headsheads, headstails, tailsheads, tailstails. If heads is the winning side, we would have the results: $729, $81, $81, $9 AVERAGE: $225
Another way to look at it is 75% of the time you lose money, 25% of the time you make money. Now, would you still take that bet?
Re: Is this a good or bad bet?
Would definitely take the bet even if it was +100% and 70%.
Since there is no mention of minimum bet size, wager can be sized to reduce risk of ruin.
My guess though is that there is some information missing as otherwise its a very simple math problem.
Since there is no mention of minimum bet size, wager can be sized to reduce risk of ruin.
My guess though is that there is some information missing as otherwise its a very simple math problem.

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Re: Is this a good or bad bet?
I'm sorry I must not be bright enough to understand all the threads regarding bets and there implied relation to the stock market. The difference is that investing in the market is not some random bet. If so then it would be up and down randomly 50% of the time. That, of course, is not true as it is positive in 2/3+ of the time. The other thing is that even though every bet is independent of the other they are LINKED together to give you a CAGR. That is why one year is MUTIPLIED to the next which is done to the next and so on. That CAGR with stock investing does go up with time. I believe the data is 50/50 in one day to 6670% in 1 yr. to near 100% in 10 yr. periods.
I'm hoping someone who is smarter then me will explain this constant analogy.
Good luck.
I'm hoping someone who is smarter then me will explain this constant analogy.
Good luck.
"The stock market [fluctuation], therefore, is noise. A giant distraction from the business of investing.”  Jack Bogle
Re: Is this a good or bad bet?
2retire wrote:No. This example is meant to teach you how losing a percent of your money is not equal to gaining that same percentage back. In order to recover from a 70% loss, you need to gain 233.33%. 170% won't cut it; you are guaranteed to lose money.
I agree. I would not take this bet.
Andy
Re: Is this a good or bad bet?
I originally read it as a 70% positive payback, but at a 170% payout it looks good. You just need to let the house make sure you make enough bets to let the odds work out
Last edited by papiper on Tue Jul 09, 2013 12:14 pm, edited 1 time in total.
Re: Is this a good or bad bet?
tadamsmar wrote:I ran across a discussion of this question in a paper by Paul Samuelson on investment management.
You have the opportunity to bet on repeated flips of a fair coin. For heads, you get a 170% gain (for instance, $100 becomes $270), for tails, you have a 70% loss (for instance, $100 becomes $30).
Would you take the bet?
Can you hedge? This "opportunity" wouldn't really exist, since in real life, someone would figure out a way to make a side bet 50/50 and take the other side, creating an arbitrage opportunity.
In any case, yes, I would take the bet. I would not let it ride, but make same amount bet each time, and do it as much as possible. One dollar at a time is better than 100.
Last edited by inbox788 on Tue Jul 09, 2013 12:03 pm, edited 1 time in total.
 nisiprius
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Re: Is this a good or bad bet?
It would depend.
Scandinavian's analysis is correct, 2retire's is not. What the example illustrates is a large positive expectation with an extremely skewed distribution of outcomes. For expectation, it is perfectly simple. There's 1/2 a chance of $270, 1/2 a chance of $30, average is $150, so on the average you multiply your money by 1.5 times on every throw. In terms of odds, probability, expectation, etc. it is overwhelmingly lucrative to play the game.
But if you parlay the game over multiple coin flips, the outcome becomes exceedingly skew and lotterylike, except that in a lottery your expectation is negative, and in this one it is positive, bigtime. There is a large chance of losing your money, and a small chance of winning, but the wins are so huge that they make up for the losses.
In order to talk more about what I would do, it's necessary to be more specific about what is really being offered. If what is being offered is a chance to make this bet, with $100, any time I likeput down the $100, flip the coin, settle up, lather, rinse, repeat, I would gladly make this bet, for $100, over and over and over again, the more times the better. I'd make it every day. I'd make it every hour.
If what is being offered is a onetime opportunity to do this with as many coinflips as I like, with results of each flip being parlayed as the start for the next bet, then things get much trickier.
P.S. Really, Scandinavian's math is perfectly simple.
HH: $100, * 2.70 = $270, * 2.7 = $729.
HT: $100, * 2.70 = $270, * 0.30 = $81.
TH: $100, * 0.30 = $30, * 2.7 = $81.
TT: $100, * 0.30 = $30, * 0.30 = $9.
Scandinavian's analysis is correct, 2retire's is not. What the example illustrates is a large positive expectation with an extremely skewed distribution of outcomes. For expectation, it is perfectly simple. There's 1/2 a chance of $270, 1/2 a chance of $30, average is $150, so on the average you multiply your money by 1.5 times on every throw. In terms of odds, probability, expectation, etc. it is overwhelmingly lucrative to play the game.
But if you parlay the game over multiple coin flips, the outcome becomes exceedingly skew and lotterylike, except that in a lottery your expectation is negative, and in this one it is positive, bigtime. There is a large chance of losing your money, and a small chance of winning, but the wins are so huge that they make up for the losses.
In order to talk more about what I would do, it's necessary to be more specific about what is really being offered. If what is being offered is a chance to make this bet, with $100, any time I likeput down the $100, flip the coin, settle up, lather, rinse, repeat, I would gladly make this bet, for $100, over and over and over again, the more times the better. I'd make it every day. I'd make it every hour.
If what is being offered is a onetime opportunity to do this with as many coinflips as I like, with results of each flip being parlayed as the start for the next bet, then things get much trickier.
P.S. Really, Scandinavian's math is perfectly simple.
HH: $100, * 2.70 = $270, * 2.7 = $729.
HT: $100, * 2.70 = $270, * 0.30 = $81.
TH: $100, * 0.30 = $30, * 2.7 = $81.
TT: $100, * 0.30 = $30, * 0.30 = $9.
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: Is this a good or bad bet?
nisiprius wrote:If what is being offered is a onetime opportunity to do this with as many coinflips as I like, with results of each flip being parlayed as the start for the next bet, then things get much trickier.
So, in this instance, how many flips would be sane? 1 flip, expect 1.5, 2 flips 2.25, etc, but win loss ratio goes to zero quickly, so 100 flips is basically guaranteed loser. Infinite flips expect infinite return, but zero chance. Even 5 is probably too many, so I'd guess somewhere between 1 to 3 flips maximizes something.
Re: Is this a good or bad bet?
I looked up the paper and it was interesting  as some have alluded here, the "average" investment over all options of coin flips is very good and increases with each additional flip over the entire range. However the number of "winners" in the game drops  at 24 bets, only 27% of the players would "win" although they would win an impressive amount. A back door argument about wealth distribution. If you could be allowed to play every outcome, you want this game. If you are only allowed one hand.... not so much.

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Re: Is this a good or bad bet?
2retire wrote:No. This example is meant to teach you how losing a percent of your money is not equal to gaining that same percentage back. In order to recover from a 70% loss, you need to gain 233.33%. 170% won't cut it; you are guaranteed to lose money.Scandinavian wrote:For example, say we flip just two times. There's four possible outcomes: headsheads, headstails, tailsheads, tailstails. If heads is the winning side, we would have the results: $729, $81, $81, $9 AVERAGE: $225
Another way to look at it is 75% of the time you lose money, 25% of the time you make money. Now, would you still take that bet?
Yes, I would. With this example (2 flips), you have 1 in 4 odds to win more than 4x your money. The expected value is positive in this case.
Would you play a game where you have a 75% chance of losing $1 and a 25% chance of making a million dollars? My point being you need to know payouts, not just odds.
Re: Is this a good or bad bet?
I wouldn't take the bet, despite the positive expected value, unless I could diversify across multiple coins and rebalance between bets.
"Essentially, all models are wrong, but some are useful."  George E. P Box

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Re: Is this a good or bad bet?
No, I'd pass on this "ground floor opportunity".
"Luck is not a strategy" Asking Portfolio Questions
Re: Is this a good or bad bet?
Betting is like market timing  avoid it.
Chaz

 “Money is better than poverty, if only for financial reasons." Woody Allen

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 Epsilon Delta
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Re: Is this a good or bad bet?
inbox788 wrote:nisiprius wrote:If what is being offered is a onetime opportunity to do this with as many coinflips as I like, with results of each flip being parlayed as the start for the next bet, then things get much trickier.
So, in this instance, how many flips would be sane? 1 flip, expect 1.5, 2 flips 2.25, etc, but win loss ratio goes to zero quickly, so 100 flips is basically guaranteed loser. Infinite flips expect infinite return, but zero chance. Even 5 is probably too many, so I'd guess somewhere between 1 to 3 flips maximizes something.
Not true. If you do 100 flips you have a 24%18% chance of winning money, and a better than 13%9% of multiplying it by more than 127. The real problem with a lot of flips is you have a real chance of breaking the house and not getting paid (1% chance of winning by a factor of 7,500,000).At a guess as you increase the number of flips the chance of winning is asymptotic to about 25% and the chance of winning by at least any particular amount is also asymptotic to the same 25%
Edit to add: Arg, I had an off by one in my probability calculation. And of course my guess turned out to be wrong, and I should have known it. I'm still happy I injected some calculations into the discussion and even happier that some others got them correct.
Last edited by Epsilon Delta on Mon Jul 15, 2013 10:14 am, edited 1 time in total.
Re: Is this a good or bad bet?
Scandinavian wrote:I'd definitely take the bet, but only with a very small piece of my net worth. The bet has a positive expected value, but you're going to lose more often than you win.
For example, say we flip just two times. There's four possible outcomes: headsheads, headstails, tailsheads, tailstails. If heads is the winning side, we would have the results: $729, $81, $81, $9 AVERAGE: $225
So, good expected value, but don't bet the farm!
If you do this enough times, you end up with something that looks like the lottery (but with better odds). You will have an exceedingly high likelihood of losing money, and an extremely tiny chance of a big payoff. Once you put the opportunity into that light, I suspect the psychological factors (utility of gains and losses) may overwhelm the purely economical aspects of the opportunity.
Best wishes.
Andy
 nisiprius
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Re: Is this a good or bad bet?
There's another interesting possibility, if the bet is a completely open offer (bet any amount any time you like as often as you like). Perhaps I am just saying the same thing as camongo is.
I'm too lazy to do the binomial distribution math, but I will say by the seat of my pants that I would be willing to bet a total of $10,000 if I could play make 10,000 separate bets for $100 each and flip the coin once. I think I would be willing to make 10,000 separate bets for $1 each, with each bet involving a threeflip parlay. I think I would be willing to bet $10,000 in the form of 1,000,000 separate bets for a penny each, with each bet being a sixflip parlay. And, of course, assuming the accounting is done to sufficiently small fractions of penny.
The bet itself become more and more skew and lotterylike as you increase the number of flips, but by doing many of them in parallel and taking the sum, the central limit theorem kicks in and undoes a lot of that skew.
In real life I wouldn't do it, though, on the grounds that anything that looks too good to be true isthat is, I wouldn't believe the person offering the bet was doing so in good faith.
I'm too lazy to do the binomial distribution math, but I will say by the seat of my pants that I would be willing to bet a total of $10,000 if I could play make 10,000 separate bets for $100 each and flip the coin once. I think I would be willing to make 10,000 separate bets for $1 each, with each bet involving a threeflip parlay. I think I would be willing to bet $10,000 in the form of 1,000,000 separate bets for a penny each, with each bet being a sixflip parlay. And, of course, assuming the accounting is done to sufficiently small fractions of penny.
The bet itself become more and more skew and lotterylike as you increase the number of flips, but by doing many of them in parallel and taking the sum, the central limit theorem kicks in and undoes a lot of that skew.
In real life I wouldn't do it, though, on the grounds that anything that looks too good to be true isthat is, I wouldn't believe the person offering the bet was doing so in good faith.
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: Is this a good or bad bet?
Mister Whale wrote:No way!
I'd be curious to see the context of the question and the subsequent discussion.
Here's the context. The question is in the section "Falsity of Corollary" on page 3 or page 2494 depending on how you are counting. It's posed as an example rather than a question, but Samuelson does ask "Isn't he [Pascal] a fool?"
http://finance.martinsewell.com/moneym ... on1971.pdf
The broader context is that I found this paper in the book: "The Kelly Capital Growth Criterion".
Last edited by tadamsmar on Tue Jul 09, 2013 2:50 pm, edited 1 time in total.
Re: Is this a good or bad bet?
staythecourse wrote:I'm sorry I must not be bright enough to understand all the threads regarding bets and there implied relation to the stock market. The difference is that investing in the market is not some random bet. If so then it would be up and down randomly 50% of the time. That, of course, is not true as it is positive in 2/3+ of the time. The other thing is that even though every bet is independent of the other they are LINKED together to give you a CAGR. That is why one year is MUTIPLIED to the next which is done to the next and so on. That CAGR with stock investing does go up with time. I believe the data is 50/50 in one day to 6670% in 1 yr. to near 100% in 10 yr. periods.
I'm hoping someone who is smarter then me will explain this constant analogy.
Good luck.
I'm sorry I must not be bright enough to understand all the posts regarding random 50% up and down bets and their implied relation to all of gambling in general.
I'm hoping someone who is smarter than me will explain this constant analogy.
Good luck.
Last edited by tadamsmar on Tue Jul 09, 2013 4:48 pm, edited 2 times in total.
Re: Is this a good or bad bet?
Cant I just not "let it ride" and repeatedly bet $100 exactly on every flip?
In that case I do it untill they eventually run out of money to pay me...
In that case I do it untill they eventually run out of money to pay me...
Re: Is this a good or bad bet?
deleted
Last edited by tadamsmar on Tue Jul 09, 2013 2:16 pm, edited 1 time in total.

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Re: Is this a good or bad bet?
Wagnerjb wrote:Scandinavian wrote:I'd definitely take the bet, but only with a very small piece of my net worth. The bet has a positive expected value, but you're going to lose more often than you win.
For example, say we flip just two times. There's four possible outcomes: headsheads, headstails, tailsheads, tailstails. If heads is the winning side, we would have the results: $729, $81, $81, $9 AVERAGE: $225
So, good expected value, but don't bet the farm!
If you do this enough times, you end up with something that looks like the lottery (but with better odds). You will have an exceedingly high likelihood of losing money, and an extremely tiny chance of a big payoff. Once you put the opportunity into that light, I suspect the psychological factors (utility of gains and losses) may overwhelm the purely economical aspects of the opportunity.
Best wishes.
I think utility (function) is the golden word here. For the $100 bet, I would say that a $170 win would be more positive than a $70 loss would be negative, for almost everyone. Change that to $1M and a $700k loss would certainly hurt more than a big gain, regardless of how big it may be!
Re: Is this a good or bad bet?
I would certainly take this bet and i am pretty sure any trader would also.
First of all, this has a positive expectancy. Then trader decides bet size to avoid "risk of ruin". One can use kelly criterion to use the optimum bet size.
http://en.wikipedia.org/wiki/Kelly_criterion
Even other wise, if one uses 2% of the capital per bet, this is huge winner even considering decent transaction costs. One can even run Montecarlo analysis to determine the max drawdown that happens in worst case scenario.
First of all, this has a positive expectancy. Then trader decides bet size to avoid "risk of ruin". One can use kelly criterion to use the optimum bet size.
http://en.wikipedia.org/wiki/Kelly_criterion
Even other wise, if one uses 2% of the capital per bet, this is huge winner even considering decent transaction costs. One can even run Montecarlo analysis to determine the max drawdown that happens in worst case scenario.
Re: Is this a good or bad bet?
As others have said, yes this is a good bet but don't parlay (i.e. bet your whole wad every time).
An even simpler (or more outrageous) version of this is: coin flip, you either get 10x your bet or you lose your entire bet. Obviously if you parlay you're essentially guaranteed to lose everything (sooner or later). But if you portion your bets it's fabulous. The aforementioned Kelly Criterion helps you to optimize your bets to make your winnings happen as fast as possible while minimizing your chance of ruin.
Interesting theoretical exercise. On the off chance it's not theoretical, please PM me (although I'd probably be with Nisiprius and back out due to it being "to good to be true".)
An even simpler (or more outrageous) version of this is: coin flip, you either get 10x your bet or you lose your entire bet. Obviously if you parlay you're essentially guaranteed to lose everything (sooner or later). But if you portion your bets it's fabulous. The aforementioned Kelly Criterion helps you to optimize your bets to make your winnings happen as fast as possible while minimizing your chance of ruin.
Interesting theoretical exercise. On the off chance it's not theoretical, please PM me (although I'd probably be with Nisiprius and back out due to it being "to good to be true".)

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Re: Is this a good or bad bet?
tadamsmar wrote:I ran across a discussion of this question in a paper by Paul Samuelson on investment management.
You have the opportunity to bet on repeated flips of a fair coin. For heads, you get a 170% gain (for instance, $100 becomes $270), for tails, you have a 70% loss (for instance, $100 becomes $30).
Would you take the bet?
It is trivial to see that this is a losing proposition. That is:
Lim (n > inf) of (2.7)^n*(0.3)^n = 0
Good way to go broke.
Best regards, Op   "In the middle of difficulty lies opportunity." Einstein
 zaboomafoozarg
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Re: Is this a good or bad bet?
(1 + 1.7) * (1  .7) = 2.7 * .3 = .81
.81 < 1
Bad bet.
.81 < 1
Bad bet.
Re: Is this a good or bad bet?
With $100,000 initial bet and an Excel random number generator. Heads you gain 170% and tails you lose 70%, after 100 flips of coin you will end up with zero.
Not a good bet.
Not a good bet.
Re: Is this a good or bad bet?
I don't have a strong background in math but I do know a bit of programming. I wrote a Python program to generate 100 outcomes of 100 coin flips, starting with $100.
If I started with $100 I'd do this because of the aforementioned lottery shot.
$ python invest.py
Run 1 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 2 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 3 completed. Terminal value is $0. Max value was $3815 at flip 28.
Run 4 completed. Terminal value is $0. Max value was $4304 at flip 6.
Run 5 completed. Terminal value is $0. Max value was $177 at flip 5.
Run 6 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 7 completed. Terminal value is $0. Max value was $1968 at flip 3.
Run 8 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 9 completed. Terminal value is $0. Max value was $35917545 at flip 35.
Run 10 completed. Terminal value is $156. Max value was $6758 at flip 33.
Run 11 completed. Terminal value is $1411. Max value was $2218531234 at flip 48.
Run 12 completed. Terminal value is $1411. Max value was $327918 at flip 90.
Run 13 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 14 completed. Terminal value is $1411. Max value was $257851 at flip 61.
Run 15 completed. Terminal value is $0. Max value was $218 at flip 3.
Run 16 completed. Terminal value is $0. Max value was $8862 at flip 20.
Run 17 completed. Terminal value is $17. Max value was $1291 at flip 7.
Run 18 completed. Terminal value is $17. Max value was $3134870 at flip 79.
Run 19 completed. Terminal value is $0. Max value was $1291 at flip 7.
Run 20 completed. Terminal value is $83352484. Max value was $1742693381 at flip 92.
Run 21 completed. Terminal value is $156. Max value was $89907 at flip 71.
Run 22 completed. Terminal value is $0. Max value was $34336 at flip 28.
Run 23 completed. Terminal value is $0. Max value was $177 at flip 5.
Run 24 completed. Terminal value is $0. Max value was $143 at flip 7.
Run 25 completed. Terminal value is $0. Max value was $18248 at flip 34.
Run 26 completed. Terminal value is $0. Max value was $250315 at flip 30.
Run 27 completed. Terminal value is $1. Max value was $7509466 at flip 29.
Run 28 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 29 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 30 completed. Terminal value is $0. Max value was $1391519 at flip 45.
Run 31 completed. Terminal value is $0. Max value was $11622 at flip 7.
Run 32 completed. Terminal value is $156. Max value was $1030105 at flip 27.
Run 33 completed. Terminal value is $1. Max value was $4304 at flip 6.
Run 34 completed. Terminal value is $17. Max value was $729 at flip 2.
Run 35 completed. Terminal value is $0. Max value was $21208 at flip 43.
Run 36 completed. Terminal value is $0. Max value was $590 at flip 4.
Run 37 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 38 completed. Terminal value is $1029043. Max value was $3329896 at flip 68.
Run 39 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 40 completed. Terminal value is $114338. Max value was $8990720 at flip 69.
Run 41 completed. Terminal value is $0. Max value was $328256 at flip 17.
Run 42 completed. Terminal value is $0. Max value was $3486 at flip 8.
Run 43 completed. Terminal value is $1411. Max value was $5814 at flip 24.
Run 44 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 45 completed. Terminal value is $0. Max value was $254 at flip 12.
Run 46 completed. Terminal value is $1. Max value was $100 at flip 0.
Run 47 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 48 completed. Terminal value is $156. Max value was $3486 at flip 8.
Run 49 completed. Terminal value is $0. Max value was $590 at flip 4.
Run 50 completed. Terminal value is $0. Max value was $2356 at flip 43.
Run 51 completed. Terminal value is $83352484. Max value was $277841613 at flip 99.
Run 52 completed. Terminal value is $0. Max value was $2287 at flip 12.
Run 53 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 54 completed. Terminal value is $156. Max value was $185110 at flip 85.
Run 55 completed. Terminal value is $0. Max value was $590 at flip 4.
Run 56 completed. Terminal value is $0. Max value was $5314 at flip 4.
Run 57 completed. Terminal value is $0. Max value was $9697 at flip 40.
Run 58 completed. Terminal value is $17. Max value was $6758 at flip 33.
Run 59 completed. Terminal value is $156. Max value was $1046 at flip 9.
Run 60 completed. Terminal value is $1. Max value was $955 at flip 62.
Run 61 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 62 completed. Terminal value is $83352484. Max value was $5808977936 at flip 91.
Run 63 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 64 completed. Terminal value is $156. Max value was $250315 at flip 30.
Run 65 completed. Terminal value is $1411. Max value was $717897 at flip 20.
Run 66 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 67 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 68 completed. Terminal value is $17. Max value was $270 at flip 1.
Run 69 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 70 completed. Terminal value is $0. Max value was $4304 at flip 6.
Run 71 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 72 completed. Terminal value is $156. Max value was $1966 at flip 76.
Run 73 completed. Terminal value is $0. Max value was $177 at flip 5.
Run 74 completed. Terminal value is $1. Max value was $449 at flip 90.
Run 75 completed. Terminal value is $0. Max value was $1014 at flip 51.
Run 76 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 77 completed. Terminal value is $1. Max value was $675851 at flip 31.
Run 78 completed. Terminal value is $0. Max value was $3282 at flip 19.
Run 79 completed. Terminal value is $0. Max value was $1968 at flip 3.
Run 80 completed. Terminal value is $1029043. Max value was $353705537 at flip 55.
Run 81 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 82 completed. Terminal value is $0. Max value was $16173 at flip 56.
Run 83 completed. Terminal value is $1411. Max value was $22528 at flip 32.
Run 84 completed. Terminal value is $1029043. Max value was $12523673 at flip 45.
Run 85 completed. Terminal value is $0. Max value was $22185312 at flip 50.
Run 86 completed. Terminal value is $17. Max value was $1291 at flip 7.
Run 87 completed. Terminal value is $0. Max value was $254 at flip 12.
Run 88 completed. Terminal value is $0. Max value was $1594 at flip 5.
Run 89 completed. Terminal value is $1029043. Max value was $36998848 at flip 66.
Run 90 completed. Terminal value is $6751551218. Max value was $8335248417 at flip 98.
Run 91 completed. Terminal value is $0. Max value was $423 at flip 28.
Run 92 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 93 completed. Terminal value is $0. Max value was $23929 at flip 21.
Run 94 completed. Terminal value is $1. Max value was $599003 at flip 53.
Run 95 completed. Terminal value is $0. Max value was $1046 at flip 9.
Run 96 completed. Terminal value is $0. Max value was $675851 at flip 31.
Run 97 completed. Terminal value is $0. Max value was $7855 at flip 42.
Run 98 completed. Terminal value is $156. Max value was $25392 at flip 83.
Run 99 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 100 completed. Terminal value is $0. Max value was $13915 at flip 47.
Python code:

If I started with $100 I'd do this because of the aforementioned lottery shot.
$ python invest.py
Run 1 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 2 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 3 completed. Terminal value is $0. Max value was $3815 at flip 28.
Run 4 completed. Terminal value is $0. Max value was $4304 at flip 6.
Run 5 completed. Terminal value is $0. Max value was $177 at flip 5.
Run 6 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 7 completed. Terminal value is $0. Max value was $1968 at flip 3.
Run 8 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 9 completed. Terminal value is $0. Max value was $35917545 at flip 35.
Run 10 completed. Terminal value is $156. Max value was $6758 at flip 33.
Run 11 completed. Terminal value is $1411. Max value was $2218531234 at flip 48.
Run 12 completed. Terminal value is $1411. Max value was $327918 at flip 90.
Run 13 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 14 completed. Terminal value is $1411. Max value was $257851 at flip 61.
Run 15 completed. Terminal value is $0. Max value was $218 at flip 3.
Run 16 completed. Terminal value is $0. Max value was $8862 at flip 20.
Run 17 completed. Terminal value is $17. Max value was $1291 at flip 7.
Run 18 completed. Terminal value is $17. Max value was $3134870 at flip 79.
Run 19 completed. Terminal value is $0. Max value was $1291 at flip 7.
Run 20 completed. Terminal value is $83352484. Max value was $1742693381 at flip 92.
Run 21 completed. Terminal value is $156. Max value was $89907 at flip 71.
Run 22 completed. Terminal value is $0. Max value was $34336 at flip 28.
Run 23 completed. Terminal value is $0. Max value was $177 at flip 5.
Run 24 completed. Terminal value is $0. Max value was $143 at flip 7.
Run 25 completed. Terminal value is $0. Max value was $18248 at flip 34.
Run 26 completed. Terminal value is $0. Max value was $250315 at flip 30.
Run 27 completed. Terminal value is $1. Max value was $7509466 at flip 29.
Run 28 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 29 completed. Terminal value is $0. Max value was $270 at flip 1.
Run 30 completed. Terminal value is $0. Max value was $1391519 at flip 45.
Run 31 completed. Terminal value is $0. Max value was $11622 at flip 7.
Run 32 completed. Terminal value is $156. Max value was $1030105 at flip 27.
Run 33 completed. Terminal value is $1. Max value was $4304 at flip 6.
Run 34 completed. Terminal value is $17. Max value was $729 at flip 2.
Run 35 completed. Terminal value is $0. Max value was $21208 at flip 43.
Run 36 completed. Terminal value is $0. Max value was $590 at flip 4.
Run 37 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 38 completed. Terminal value is $1029043. Max value was $3329896 at flip 68.
Run 39 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 40 completed. Terminal value is $114338. Max value was $8990720 at flip 69.
Run 41 completed. Terminal value is $0. Max value was $328256 at flip 17.
Run 42 completed. Terminal value is $0. Max value was $3486 at flip 8.
Run 43 completed. Terminal value is $1411. Max value was $5814 at flip 24.
Run 44 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 45 completed. Terminal value is $0. Max value was $254 at flip 12.
Run 46 completed. Terminal value is $1. Max value was $100 at flip 0.
Run 47 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 48 completed. Terminal value is $156. Max value was $3486 at flip 8.
Run 49 completed. Terminal value is $0. Max value was $590 at flip 4.
Run 50 completed. Terminal value is $0. Max value was $2356 at flip 43.
Run 51 completed. Terminal value is $83352484. Max value was $277841613 at flip 99.
Run 52 completed. Terminal value is $0. Max value was $2287 at flip 12.
Run 53 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 54 completed. Terminal value is $156. Max value was $185110 at flip 85.
Run 55 completed. Terminal value is $0. Max value was $590 at flip 4.
Run 56 completed. Terminal value is $0. Max value was $5314 at flip 4.
Run 57 completed. Terminal value is $0. Max value was $9697 at flip 40.
Run 58 completed. Terminal value is $17. Max value was $6758 at flip 33.
Run 59 completed. Terminal value is $156. Max value was $1046 at flip 9.
Run 60 completed. Terminal value is $1. Max value was $955 at flip 62.
Run 61 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 62 completed. Terminal value is $83352484. Max value was $5808977936 at flip 91.
Run 63 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 64 completed. Terminal value is $156. Max value was $250315 at flip 30.
Run 65 completed. Terminal value is $1411. Max value was $717897 at flip 20.
Run 66 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 67 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 68 completed. Terminal value is $17. Max value was $270 at flip 1.
Run 69 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 70 completed. Terminal value is $0. Max value was $4304 at flip 6.
Run 71 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 72 completed. Terminal value is $156. Max value was $1966 at flip 76.
Run 73 completed. Terminal value is $0. Max value was $177 at flip 5.
Run 74 completed. Terminal value is $1. Max value was $449 at flip 90.
Run 75 completed. Terminal value is $0. Max value was $1014 at flip 51.
Run 76 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 77 completed. Terminal value is $1. Max value was $675851 at flip 31.
Run 78 completed. Terminal value is $0. Max value was $3282 at flip 19.
Run 79 completed. Terminal value is $0. Max value was $1968 at flip 3.
Run 80 completed. Terminal value is $1029043. Max value was $353705537 at flip 55.
Run 81 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 82 completed. Terminal value is $0. Max value was $16173 at flip 56.
Run 83 completed. Terminal value is $1411. Max value was $22528 at flip 32.
Run 84 completed. Terminal value is $1029043. Max value was $12523673 at flip 45.
Run 85 completed. Terminal value is $0. Max value was $22185312 at flip 50.
Run 86 completed. Terminal value is $17. Max value was $1291 at flip 7.
Run 87 completed. Terminal value is $0. Max value was $254 at flip 12.
Run 88 completed. Terminal value is $0. Max value was $1594 at flip 5.
Run 89 completed. Terminal value is $1029043. Max value was $36998848 at flip 66.
Run 90 completed. Terminal value is $6751551218. Max value was $8335248417 at flip 98.
Run 91 completed. Terminal value is $0. Max value was $423 at flip 28.
Run 92 completed. Terminal value is $0. Max value was $729 at flip 2.
Run 93 completed. Terminal value is $0. Max value was $23929 at flip 21.
Run 94 completed. Terminal value is $1. Max value was $599003 at flip 53.
Run 95 completed. Terminal value is $0. Max value was $1046 at flip 9.
Run 96 completed. Terminal value is $0. Max value was $675851 at flip 31.
Run 97 completed. Terminal value is $0. Max value was $7855 at flip 42.
Run 98 completed. Terminal value is $156. Max value was $25392 at flip 83.
Run 99 completed. Terminal value is $0. Max value was $100 at flip 0.
Run 100 completed. Terminal value is $0. Max value was $13915 at flip 47.
Python code:

Code: Select all
import random
total_flips = 100
# flips = 0
# money = 100
# current_flip = 0
total_runs = 100
run = 0
while (run < total_runs):
run += 1
flip = 0
money = 100
current_flip = 0
max_this_run = (0, 100)
while (flip < total_flips):
flip += 1
current_flip = random.randrange(2)
if current_flip == 0:
money = money * 0.3
else:
money = money * 2.7
if max_this_run[1] < money:
max_this_run = (flip, money)
print "Run %s completed. Terminal value is $%s. Max value was $%s at flip %s." % (run, int(money), int(max_this_run[1]), max_this_run[0])
Re: Is this a good or bad bet?
I find it interesting that a few posters seem to think there is some sort of clear distinction between betting/gambling and investing. I think one who believes that can still be a good investor, since "cookbook" investing (investing based on standardized advice) is good investing these days.
The legal/regulatory basis for the distinction between gambling and investing is based on the fact that a subset of betting activities has the side effect of providing liquidity for trade, business, the economy. This liquidity is needed, so certain types of betting are legal. Betting that does not have this side effect is illegal or more strictly regulated, probably because it causes social disruption due to the power of random reinforcement schedules on behavior, particularly among some prone to compulsive gambling. But the two side effects can mix, betting that provides liquidity can lead to compulsive behavior.
The mathematics of estimating return and risk are the same. Both can be profitable activities if the expectation of return is positive and the risks are managed effectively.
The legal/regulatory basis for the distinction between gambling and investing is based on the fact that a subset of betting activities has the side effect of providing liquidity for trade, business, the economy. This liquidity is needed, so certain types of betting are legal. Betting that does not have this side effect is illegal or more strictly regulated, probably because it causes social disruption due to the power of random reinforcement schedules on behavior, particularly among some prone to compulsive gambling. But the two side effects can mix, betting that provides liquidity can lead to compulsive behavior.
The mathematics of estimating return and risk are the same. Both can be profitable activities if the expectation of return is positive and the risks are managed effectively.
Re: Is this a good or bad bet?
Epsilon Delta wrote:inbox788 wrote:nisiprius wrote:If what is being offered is a onetime opportunity to do this with as many coinflips as I like, with results of each flip being parlayed as the start for the next bet, then things get much trickier.
So, in this instance, how many flips would be sane? 1 flip, expect 1.5, 2 flips 2.25, etc, but win loss ratio goes to zero quickly, so 100 flips is basically guaranteed loser. Infinite flips expect infinite return, but zero chance. Even 5 is probably too many, so I'd guess somewhere between 1 to 3 flips maximizes something.
Not true. If you do 100 flips you have a 24% chance of winning money, and a better than 13% of multiplying it by more than 127. The real problem with a lot of flips is you have a real chance of breaking the house and not getting paid (1% chance of winning by a factor of 7,500,000). At a guess as you increase the number of flips the chance of winning is asymptotic to about 25% and the chance of winning by at least any particular amount is also asymptotic to the same 25%
You are correct. My mistake was thinking there was a decaying win rate. With the positive expected, even if you lose a few flips, you still have a 25% chance of coming back. You can get ahead quite a bit and outcomes are still unknown.
Run 9 completed. Terminal value is $0. Max value was $35917545 at flip 35.
Run 18 completed. Terminal value is $17. Max value was $3134870 at flip 79.
Run 27 completed. Terminal value is $1. Max value was $7509466 at flip 29.
Run 80 completed. Terminal value is $1029043. Max value was $353705537 at flip 55.
So does the optimal strategy become flip as many times as it takes to break the bank with 25% likelihood? Or is there a rational reason to pick another minimum or maximum number of flips?
Re: Is this a good or bad bet?
I still don't see the downside of taking my entire networth, whatever that may be, dividing it in to the smallest allowable betsize possible, say 1 penny, make (networth/0.01) number of bets, and choosing to flip once for each bet (networth/0.01 number of total flips).
Re: Is this a good or bad bet?
Epsilon Delta wrote:Not true. If you do 100 flips you have a 24% chance of winning money, and a better than 13% of multiplying it by more than 127.
How are you calculating this? I get lower probabilities...but my math could be wrong.
In 100 flips, assuming you parlay the winnings, i think you need at least 55 winning flips to come out ahead.
2.7^55*0.3^45 = 1.57
while with 54 wins or fewer you lose money:
2.7^54*0.3^46 = 0.17
The probability of getting at least 55 wins can be calculated in Excel with:
=1BINOMDIST(54,100,0.5,1) = 18.4%
Multiplying money by 127 or more requires at least 57 winning flips:
2.7^57*0.3^43 = 127.04
The probability of at least 57 winning flips is:
=1BINOMDIST(56,100,0.5,1) = 9.6%
Last edited by camontgo on Wed Jul 10, 2013 2:24 pm, edited 2 times in total.
"Essentially, all models are wrong, but some are useful."  George E. P Box
Re: Is this a good or bad bet?
amoeba wrote:I still don't see the downside of taking my entire networth, whatever that may be, dividing it in to the smallest allowable betsize possible, say 1 penny, make (networth/0.01) number of bets, and choosing to flip once for each bet (networth/0.01 number of total flips).
There isn't a downside, but there also isn't the possibility of spectacular wealth by arranging it in parallel instead of serially.
Your outcome is predictable in that you'll turn $100K into $150K:
10000000 flips completed. New net worth is $149941. Heads: 5002417, Tails: 4997583
10000000 flips completed. New net worth is $150056. Heads: 4997657, Tails: 5002343
10000000 flips completed. New net worth is $150011. Heads: 4999511, Tails: 5000489
Code: Select all
import random
networth = 100000
pennies = networth * 100 # net worth in pennies
flip = 0
new_networth = 0
heads = 0
tails = 0
while (flip < pennies):
flip += 1
current_flip = random.randrange(2)
if current_flip == 0:
new_networth = new_networth + (0.01 * 0.3)
heads += 1
else:
new_networth = new_networth + (0.01 * 2.7)
tails += 1
print "%s flips completed. New net worth is $%s. Heads: %s, Tails: %s" % (flip, int(new_networth), heads, tails)
Re: Is this a good or bad bet?
delete.
oops. I read the question wrong. I thought $270 when profit and $70 when loss.
Carry on..
oops. I read the question wrong. I thought $270 when profit and $70 when loss.
Carry on..
Last edited by Ranger on Wed Jul 10, 2013 2:54 pm, edited 2 times in total.
Re: Is this a good or bad bet?
bigred77 wrote:Cant I just not "let it ride" and repeatedly bet $100 exactly on every flip?
In that case I do it untill they eventually run out of money to pay me...
I agree with the above.
In investing, I would advocate utilizing annualized/geometric mean returns that would give an expected 10% return per event. However, in the betting context with discreet events, utilizing consistent amounts and an average/arithmetic mean would yield an expected 50% return per event. [If investing and letting it ride, I'd imagine rebalancing after every event would increase expected return]
I'm sure there is an optimal method of escalating the amount of each subsequent bet that would enable fabulous wealth growth, but that doesn't seem to be the point of the example. The St. Petersburg Paradox is also an interesting in the invest v. bet paradigm, with the outcomes presumably reversing for the two.
Re: Is this a good or bad bet?
I want to use this Kelly Calculator, but first I need to cast the problem in the correct terms for the calculator input:
http://www.albionresearch.com/kelly/default.php
Since you only lose 70% (0.7), 30% of your bet is not at risk.
You get either 0.7 or 2.4 times the amount bet.
The odds offered are 1.7:0.7 or 17:7
(if the odds offered are a:b then you get b for a loss and a+b for a win.)
The probability of winning is 50%
Plugging this in to the calculator, you see that betting 29.41% of your capital gives you the maximum log growth rate.
But, you need repeatedly bet 0.3+0.2941 = %59.41 of your capital to include the 30% of the bet that is not at risk to maximize log growth.
Samuelson presented this as an all or nothing proposition. He claimed that Pascal would take the bet based on a utility function other than log growth. I am not sure if I believe him since I am not sure Pascal was not aware of capital management strategies that postdate him, but I have not dug too deeply into the utility function argument. Here's the Samuelson paper:
http://finance.martinsewell.com/moneym ... on1971.pdf
You also take a big risk to maximize long term log growth, a 50% risk of losing about 20% of your capital on each bet. If this represented the yearwise risk of a retirement investor, then the risk would be intolerable. Even young investors typically don't (jn effect) bet more than about half the Kelly fraction of capital by going all in on the stock market (assuming the analogy to the stock market is even valid).
http://www.albionresearch.com/kelly/default.php
Since you only lose 70% (0.7), 30% of your bet is not at risk.
You get either 0.7 or 2.4 times the amount bet.
The odds offered are 1.7:0.7 or 17:7
(if the odds offered are a:b then you get b for a loss and a+b for a win.)
The probability of winning is 50%
Plugging this in to the calculator, you see that betting 29.41% of your capital gives you the maximum log growth rate.
But, you need repeatedly bet 0.3+0.2941 = %59.41 of your capital to include the 30% of the bet that is not at risk to maximize log growth.
Samuelson presented this as an all or nothing proposition. He claimed that Pascal would take the bet based on a utility function other than log growth. I am not sure if I believe him since I am not sure Pascal was not aware of capital management strategies that postdate him, but I have not dug too deeply into the utility function argument. Here's the Samuelson paper:
http://finance.martinsewell.com/moneym ... on1971.pdf
You also take a big risk to maximize long term log growth, a 50% risk of losing about 20% of your capital on each bet. If this represented the yearwise risk of a retirement investor, then the risk would be intolerable. Even young investors typically don't (jn effect) bet more than about half the Kelly fraction of capital by going all in on the stock market (assuming the analogy to the stock market is even valid).
Re: Is this a good or bad bet?
Absolutely take this. Some have done the math correctly, some have not.
Re: Is this a good or bad bet?
I did two flips:
First flip: Postive $270
Second Flip negative: $81
Long term, this is a losing proposition, just based on probability.
While long term positives theoretically possible, the odds are strongly against it.
You would have a 50/50 chance if the second one was 62% loss
you would have almost even chances. (99.9 was the outcome from two tosses).
First flip: Postive $270
Second Flip negative: $81
Long term, this is a losing proposition, just based on probability.
While long term positives theoretically possible, the odds are strongly against it.
You would have a 50/50 chance if the second one was 62% loss
you would have almost even chances. (99.9 was the outcome from two tosses).
Re: Is this a good or bad bet?
If you invest all your capital in a unbounded series of bets your outcome is a random series of heads and tails. In the long run there will be half heads and half tails. It will be a long multiplicative series of 0.3 and 2.7. you can arrange the series to be:
HTHTHT...
or
2.7*0.7*2.7*0.7*2.7*0.7*...
2.7*0.7 = 0.81 = 0.9*0.9
So, the series amounts to: 0.9*0.9*.... That is, you lose 10% on each bet on average.
Now, if you invest half your available capital on each round, the series is:
(0.5+0.5*2.7)*(0.5+0.5*0.3)*(0.5+0.5*2.7)*(0.5+0.5*0.3)*...
1.85*0.65*1.85*0.65*...
or:
1.2025*1.2025*... That is, your capital grows over 20% per each 2 consecutive flips on average!
Now, if you replace 0.5 with F and optimize F for the growth rate, then F = the Kelly fraction, the fraction of your capital that provides the highest growth rate over the long run.
HTHTHT...
or
2.7*0.7*2.7*0.7*2.7*0.7*...
2.7*0.7 = 0.81 = 0.9*0.9
So, the series amounts to: 0.9*0.9*.... That is, you lose 10% on each bet on average.
Now, if you invest half your available capital on each round, the series is:
(0.5+0.5*2.7)*(0.5+0.5*0.3)*(0.5+0.5*2.7)*(0.5+0.5*0.3)*...
1.85*0.65*1.85*0.65*...
or:
1.2025*1.2025*... That is, your capital grows over 20% per each 2 consecutive flips on average!
Now, if you replace 0.5 with F and optimize F for the growth rate, then F = the Kelly fraction, the fraction of your capital that provides the highest growth rate over the long run.
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