# Is it ever possible to "win" a 50/50 game with a clear goal in mind.

Mathematics Asked by Big Altz on January 3, 2022

Lets say I have 750 dollars and want at least 1250 dollars at the end of a 50/50 game where I can bet any possible value. Is there any way in which I can raise my chances of winning? And if there is, how can I define the best initial bet to be doubled?

(At first I thought problems like these were easy to solve and there was no possible way to "win", but at the same time I have some doubts about it and don’t know the mathematical explanation for it.)

Your optimal probability of getting to $1250 from$750 with fair bets is 60% and you cannot do better. One strategy of doing so is given by @Robert's answer above, called "bold play" in the literature. If you follow his strategy, you find yourself along the following tree: You take each path of this betting tree with probability $$1/2$$. If your initial probability of winning from $750 is x, then one of three things happen on your way to victory: 1. You win instantly ($$50%$$ chance) 2. After an initial loss, you win 3x in a row ($$1/16 = 6.25%$$ chance) 3. After an initial loss and 2 wins, you lose again and get back to your starting fortune of$750 ($$1/16 = 6.25%$$ chance)

So we have the equation $$x = 0.5 + 0.0625 + 0.0625x,$$ and when solved, we learn that $$x = 9/16 + x/16$$, or $$15x/16 = 9/16$$, whence $$x = 9/15 = 0.6$$. (The other $$40%$$ probability is that you go broke.)

Why can't we do better than $$60%$$? Consider your fortune $$W_n$$ on the $$n$$th bet, and imagine you make the bet $$A_n$$ subject to the constraints $$A_n = 0$$ if $$W_n geq 1250$$ or $$W_n = 0$$, $$0 leq A_n leq W_n$$. Then $$W_n$$ is a uniformly integrable martingale, because your winning probability on each bet is $$1/2$$ and $$0 leq W_n leq 2*(1250-1) = 2498$$ is a bounded sequence above and below. This means that $$W_n$$ converges almost surely and in $$L^1$$ to a bounded random variable $$W_infty$$, representing your limiting winnings. The $$L^1$$ convergence is crucial here, as it means $$Bbb{E}[W_infty] = 750.$$

You have "won" your betting sequence if $$W_infty geq 1250$$. What is the probability of that? By Markov's inequality, $$Bbb{P}(W_infty geq 1250) leq frac{Bbb{E}[W_infty]}{1250} leq frac{750}{1250} = 60%.$$

(If you started with $$x$$ dollars instead, the same argument shows you have an $$x/1250$$ chance of getting to $1250, one way to do this being bold play.) So bold play actually gives you the "best possible" chance of reaching$1250--although it's also "worst possible" among strategies that achieve $$60%$$, as the other $$40%$$ of your time you lose all of your initial $750. Other betting strategies than bold play may leave you with a better probability of a nonzero limiting fortune $$W_infty$$ (one example: bet $$1/(n+1)$$ of your current wealth $$W_n$$ on bet number $$n$$ or $$1250 - W_n$$, whichever is smaller). But no strategy, however clever or intricate, can do better than a $$60%$$ chance of reaching$1250 from an initial stake of $750. Answered by Rivers McForge on January 3, 2022 By "50/50 game" I assume you mean you can make any sequence of fair bets at $$50/50$$ odds, as long as you never bet more than you have at the time. Since these are fair bets, your expected value at the end is the same as your initial fortune, $$750$$. As long as you ensure that the only possible final outcomes are $$0$$ and $$1250$$, you maximize the probability of ending with $$1250$$. Thus one possible strategy is: whenever you have $$x$$ where $$0 < x < 1250$$, bet the minimum of $$x$$ and $$1250 - x$$. With probability $$1$$, you will eventually end up at either $$1250$$ or $$0$$. Answered by Robert Israel on January 3, 2022 ## Add your own answers! ## Related Questions ### How do you take the derivative$frac{d}{dx} int_a^x f(x,t) dt$? 1 Asked on December 1, 2021 by klein4 ### Uniqueness of measures related to the Stieltjes transforms 1 Asked on December 1, 2021 ### Does there exist a function which is real-valued, non-negative and bandlimited? 1 Asked on December 1, 2021 by muzi ### Suppose$A , B , C$are arbitrary sets and we know that$( A times B ) cap ( C times D ) = emptyset$What conclusion can we draw? 3 Asked on November 29, 2021 by anonymous-molecule ### Can a nonsingular matrix be column-permuted so that the diagonal blocks are nonsingular? 1 Asked on November 29, 2021 by syeh_106 ### Does${f(x)=ln(e^{x^2})}$reduce to${x^2ln(e)}$or${2xln(e)}$? 3 Asked on November 29, 2021 by evo ### Proving that$(0,1)$is uncountable 2 Asked on November 29, 2021 by henry-brown ### Understanding a statement about composite linear maps 1 Asked on November 29, 2021 ### Assert the range of a binomial coefficient divided by power of a number 3 Asked on November 29, 2021 by vib_29 ### What is the Fourier transform of$|x|$? 3 Asked on November 29, 2021 ### Exists$t^*in mathbb{R}$such that$y(t^*)=-1$?. 2 Asked on November 29, 2021 by user514695 ### Proving$logleft(frac{4^n}{sqrt{2n+1}{2nchoose n+m}}right)geq frac{m^2}{n}$2 Asked on November 29, 2021 by zaragosa ### Linearized system for$ begin{cases} frac{d}{dt} x_1 = -x_1 + x_2 \ frac{d}{dt} x_2 = x_1 – x_2^3 end{cases} $is not resting at rest point? 1 Asked on November 29, 2021 by user3137490 ### Is it possible to construct a continuous and bijective map from$mathbb{R}^n$to$[0,1]$? 3 Asked on November 29, 2021 by kaaatata ### If$lim_{ntoinfty}|a_{n+1}/a_n|=L$, then$lim_{ntoinfty}|a_n|^{1/n}=L$2 Asked on November 29, 2021 by diiiiiklllllll ### Right adjoint to the forgetful functor$text{Ob}$1 Asked on November 29, 2021 by alf262 ### Always factorise polynomials 1 Asked on November 29, 2021 by beblunt ### Formulas for the Spinor Representation Product Decompositions$2^{[frac{N-1}{2}]} otimes 2^{[frac{N-1}{2}]}=?\$ and …

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