# Mathematical paradox? **URL:** \ **Category:** Factual Questions\ **Created:** [July 30, 2014, 2:08am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387 "2014-07-30T02:08:19Z")\ **Posts on this page:** 16\ **Page:** 5
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 31, 2014, 2:06am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/81 "2014-07-31T02:06:46Z")
> [@Bryan\_Ekers](#): > > Even if the bankroll is reduced to a mere $10, the odds of busting within 60 trials is about .001 and gets smaller as the game continues, though the online Poisson calculator I used lacks the necessary precision. For fun, I decided to calculate this exactly. The probability of busting on a bankroll of $10 within 60 trials is 39464863030417006100870577/3^60, which is .000930969… . Ugly Haskell below (which solves this problem easily, but is not at all what one would use to efficiently solve it for much larger numbers of trials, should that need ever arise for you): ```auto advance (busted:otherprobs) = let ups = (busted:0:otherprobs) downs = (busted + head otherprobs):((tail otherprobs)++[0, 0]) in zipWith (\a b -> 2 * a + 1 * b) ups downs bustProb bankroll numTurns = let initialDist = [if n < bankroll then 0 else 1 | n <- [0..bankroll]] numer:_ = (iterate advance initialDist) !! numTurns denom = 3^numTurns in (numer, denom) ``` ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 31, 2014, 2:24am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/82 "2014-07-31T02:24:43Z")
> [@Bryan\_Ekers](#): > > The player’s holdings at any given moment seem to consistently approximate one-third of the number of games played: Of course: With a 2/3 chance of gaining $1 and a 1/3 chance of losing $1 on each flip, we have a probabilistic average single-flip profit of 2/3 \* $1 - 1/3 \* $1 = $1/3. And the number of flips you are taking is gargantuan relative to the variance per flip, so the overall profit over the number of flips is very unlikely to stray significantly from this. But the computer is only doing in its simulations an imperfect approximation of something you might just as well do perfectly by hand! That’s always what’s weird to me about people doing these sorts of things, though I guess it’s at least more fun here than when people use it to convince themselves of Monty Hall or what have you. But different strokes for different folks. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 31, 2014, 2:32am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/83 "2014-07-31T02:32:01Z")
And I suppose “imperfect approximation” presupposes a particular notion of what it is the goal of such simulation is; your goal might just be the entertainment of watching the dice roll, so to speak, especially in a situation like this where sampling one point from the distribution is potentially easier to grok than writing out the full thing. So, really, that last paragraph of mine should be rewritten or stricken… Please forget I said it. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 31, 2014, 2:33am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/84 "2014-07-31T02:33:07Z")
I’m not certain if you realize this, or if it’s intentional, or even if it’s appropriate for me to so comment, but your posts occasionally seem to carry a condescending tone. ---
**Author:** ![wolfpup](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolfpup/32/10618_2.png) [@wolfpup](https://boards.straightdope.com/u/wolfpup)\ **Post date:** [July 31, 2014, 2:37am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/85 "2014-07-31T02:37:19Z")
> [@Bryan\_Ekers](#): > > There’s an on-point chapter in Martin Gardner’s _Mathematical Circus_ (which I highly recommend) called “Random Walks and Gambling”. Among other examples, it describes a coin-toss game between players A and B similar the OP’s proposal in which heads means B gives A a dollar tails means he loses A gives B a dollar. If each of the players start with a finite bankroll, the odds of either wining at any given moment is equal to their share of the total stake, i.e. if A has $7 and B has $10, A’s chances are 7/17 and B’s are 10/17. Garner adds that if one player has an infinite bankroll, the one with the finite stake is “almost certain” to face “eventual ruin”. > > **This example is based on even odds with each event, though, not the biased coin the OP proposes.** Given that advantage, I suppose it’s possible that if the game continues long enough, the player is more likely to win _every dollar on Earth_ before he goes bust. The part that I bolded is key. Analogy: the average person who walks into a Vegas casino is likely to leave with less money than he went in with. The longer he’s in there, the more likely it is that the casino’s odds will work against him and the more he’s going to lose. As his time in the casino tends toward infinity, his financial outcome tends toward the inevitable loss of everything. > [@Chronos](#): > > It’s not _quite_ the same for infinity tosses as it is for a large finite number of tosses, since no matter how far up you go, there’s always a nonzero chance that you’ll still lose it. However, as the finite number of tosses goes up, the probability approaches the probability for an infinite number of tosses, and the convergence is extremely rapid, so saying they’re the same is a very good approximation. Agreed. The central point here is that in the described situation the probability of the outcome at infinity is – far from being some meaningless unknown – actually very precisely defined. It’s like my previous analogy of integral calculus: it finds the area under a curve by summing the areas of an infinite number of infinitesimal rectangles, which turns out to be – not a meaningless number or “infinity” – but a precise measure of that exact area! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 31, 2014, 2:37am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/86 "2014-07-31T02:37:23Z")
> [@Bryan\_Ekers](#): > > I’m not certain if you realize this, or if it’s intentional, or even if it’s appropriate for me to so comment, but your posts occasionally seem to carry a condescending tone. Yes, I did realize that, and that’s why I wished to apologize for that aspect of my post. It’s not intentional, but it’s certainly appropriate for you to mention it. I shall try to be better about avoiding that. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 31, 2014, 3:02am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/87 "2014-07-31T03:02:50Z")
Hmm, on review I can see that I made quite the hash in my post 77 description of Gardner’s essay, no doubt from first trying to describe it from memory, than taking the trouble of finding the book for a more exact description, then sloppily merging the two. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 31, 2014, 3:07am UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/88 "2014-07-31T03:07:45Z")
_Ack, never mind_ ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 31, 2014, 3:00pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/89 "2014-07-31T15:00:51Z")
For what it’s worth, I don’t see **Indistinguishable** as condescending: I see him as being obsessed with precise language, terminology, and expression, together with a bit of frustration at people not saying quite exactly what they mean. Which is a worthwhile trait in a mathematician, but I suppose can seem a bit harsh to a layman. ---
**Author:** ![jtur88](https://avatars.discourse-cdn.com/v4/letter/j/e9c0ed/32.png) [@jtur88](https://boards.straightdope.com/u/jtur88)\ **Post date:** [July 31, 2014, 3:35pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/90 "2014-07-31T15:35:01Z")
> [@Bryan\_Ekers](#): > > Gardner adds that if one player [in a coin toss game] has an infinite bankroll, the one with the finite stake is “almost certain” to face “eventual ruin”. > . I have, frustratingly, argued that a casino would still win overall, even if every game were even and there was no house advantage in the per-play odds. Simply because, as a practical real-world matter, the casino has an “infinite” amount of money, and can play “forever” without going busted. Gardner appears to support that position. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [July 31, 2014, 3:48pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/91 "2014-07-31T15:48:00Z")
> [@jtur88](#): > > I have, frustratingly, argued that a casino would still win overall, even if every game were even and there was no house advantage in the per-play odds. Simply because, as a practical real-world matter, the casino has an “infinite” amount of money, and can play “forever” without going busted. Gardner appears to support that position. Who has said otherwise? ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 31, 2014, 5:24pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/92 "2014-07-31T17:24:10Z")
Sure, but with a bias that is two to one in your favour, you’d be a fool NOT to play until you’d amassed a large enough pile that just sticking it in the bank at 1% gives you a faster return than playing for a buck at a time. ---
**Author:** ![jtur88](https://avatars.discourse-cdn.com/v4/letter/j/e9c0ed/32.png) [@jtur88](https://boards.straightdope.com/u/jtur88)\ **Post date:** [July 31, 2014, 6:06pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/93 "2014-07-31T18:06:53Z")
> [@The\_Great\_Unwashed](#): > > Who has said otherwise? I don’t have the energy to search for it, but I believe I was shouted down in this very forum for such a preposterous position in a different thread… ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 31, 2014, 6:16pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/94 "2014-07-31T18:16:39Z")
The casino might have a much larger bankroll than any individual player, but do they have a larger bankroll than all of the players combined? ---
**Author:** ![Bones\_Daley](https://avatars.discourse-cdn.com/v4/letter/b/bc8723/32.png) [@Bones\_Daley](https://boards.straightdope.com/u/Bones_Daley)\ **Post date:** [July 31, 2014, 6:54pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/95 "2014-07-31T18:54:10Z")
> [@Chronos](#): > > The casino might have a much larger bankroll than any individual player, but do they have a larger bankroll than all of the players combined? Probably not, but what they _ **do** _ have (and very wisely) is a limit to the amount which any individual player can wager on any particular event. In Horace Levinson’s excellent tome “The Science of Chance” he recounts an anecdote wherein one of the Rothschild dynasty says to M. Blanc (the owner of the casino at Monte Carlo) “Take off your limits and I will play all day and all night”. M.Blanc very understandably refused his offer. ---
**Author:** ![wolfpup](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolfpup/32/10618_2.png) [@wolfpup](https://boards.straightdope.com/u/wolfpup)\ **Post date:** [July 31, 2014, 7:52pm UTC](https://boards.straightdope.com/t/mathematical-paradox/694387/96 "2014-07-31T19:52:10Z")
> [@jtur88](#): > > I have, frustratingly, argued that a casino would still win overall, even if every game were even and there was no house advantage in the per-play odds. Simply because, as a practical real-world matter, the casino has an “infinite” amount of money, and can play “forever” without going busted. Gardner appears to support that position. I disagree. Gardner’s claim depends on playing until one player goes bust. That’s not how casino games generally work in real life. The game ends when a player feels like it, or it’s time to go home – and this is determined by the player, not the casino. Moreover, to paraphrase what **Chronos** said, while the casino effectively has an infinite amount of money, the world effectively provides an infinite number of players. If every game were even, over the long term the casino would probably just break even and would not be able to pay its bills. > [@Bones\_Daley](#): > > Probably not, but what they _ **do** _ have (and very wisely) is a limit to the amount which any individual player can wager on any particular event. Simply because in any single instance, the outcome might defy the odds. [Previous page](https://boards.straightdope.com/t/mathematical-paradox/694387.md?page=4)