# A probability video game

Inspiration: A close friend asked me this inquiry.

The Problem: Intend you start with a buck. You turn a reasonable coin, if it come down on heads you win $50$ cents or else you shed $50$ cents. If after $n$ turns you have a nonzero quantity of loan, you win. What's the probability you win? What concerning the restricting instance as $n$ often tends to infinity?

modify: In this video game you are not permitted to have adverse loan. Many Thanks, Jonathan Fischoff, the connected aided substantially.

Here is an entirely probabilistic evidence motivated by Moron in Finding a clever solution to a game of chance.

Allow p be the probability that I wind up with a bottom line of 50 cents, after that $p = \frac{1}{2} + \frac{1}{2} p^3$. That is to claim, I either shed 50 cents, or obtain a buck and also shed a buck and also 50 cents at some point. Addressing for p I get 1, $(-1-\sqrt{5})/2$, and also $(1-\sqrt{5})/2$ and also it is not tough to see that we can disregard the first 2. Currently the probability I wind up with a bottom line of one buck is $p^2 = (3-\sqrt{5})/2$.

As Issac mentioned, this is called the Gamler's wreck trouble. I lately created a number of post (this and also the blog post connected from there) clarifying just how to compute the wreck probability. I'll duplicate component of among the evidence below.

**Trouble Formulation **

A casino player gets in an online casino with $n$ bucks in cash money and also begins playing a video game where he wins with probability $p$ and also looses with probability $q = 1-p$ The gampler plays the video game repetitively, wagering $1$ buck in each round. He leaves the offered it his complete lot of money gets to $N$ or he lacks loan (he is

wrecked), whichever takes place first. What is the probability that the casino player is wrecked.

A casino player's wreck can be designed as a one - dimensional arbitrary stroll in which we want the striking probability of the soaking up states. Computing these chances is rather uncomplicated. Allow $P_N(n)$ represent the probability that the casino player's lot of money gets to $N$ bucks prior to he is wrecked on the problem that his existing lot of money is $n$. After that,

$P_N(n) = p P_N(n+1) + q P_N(n-1)$

which can be revised as

$\displaystyle [P_N(n+1) - P_N(n)] = \left(\frac q p \right)[ P_N(n) - P_N(n-1)]$

Since $P_N(0) = 0$, we have that

$\displaystyle P_N(2) - P_N(1) = \left(\frac qp \right) P_N(1)$

and also in a similar way

$\displaystyle P_N(3) - P_N(2) = \left(\frac qp \right) [P_N(2) - P_N(1)] = \left( \frac qp \right)^2 P_N(1)$

Continuing in this manner, we get that

$\displaystyle P_N(n) - P_N(n-1) = \left( \frac qp \right)^{n-1} P_N(1) $.

and also consequently, by including the first $n$ such terms, we get

$\displaystyle P_N(n) = \sum_{k=0}^{n-1} \left( \frac qp \right)^k P_N(1)$.

In addition, we understand that

$\displaystyle P_N(N) = \sum_{k=0}^{N-1} \left( \frac qp \right)^k P_N(1) = 1$.

Hence,

$\displaystyle P_N(1) = \frac 1{\sum_{k=0}^{N-1} \left( \frac qp \right)^k} = \frac { 1 - (q/p)}{\strut 1 - (q/p)^N}, \quad p \neq q $

$P_N(1) = \frac 1N, \quad p = q $.

Incorporating with the previous expression for $P_N(n)$ we get,

$\displaystyle P_N(n) = \begin{cases} \frac{ 1 - (q/p)^n} {\strut 1 - (q/p)^N}, & p \neq 1/2 \ \frac{n}{N}, & p = 1/2 \end{cases}$.

For convenience of depiction allowed $\lambda = q/p$. After that, the probability of *winning * are

$\displaystyle P_N(n) = \frac{ 1 - \lambda^n} {\strut 1 - \lambda^N}, \quad \lambda \neq 1 $

$P_N(n) = \frac{n}{N}, \quad \lambda = 1 $.

As n mosts likely to infinity, you lack loan with probability 1. This is called gambler's ruin or the intoxicated stroll - - if you are winning/losing loan or strolling side to side such that you win/lose/go left/go right with probability 1/2 (a 1 - d straightforward symmetrical random walk), and also there is a border like going damaged or winding up in the seamless gutter, as the variety of games/steps mosts likely to infinity, the probability of striking that border mosts likely to 1.

I do not recognize just how to confirm it, yet per Joshua Zucker's talk about A001405 in the OEIS, and also concurring with T.'s solution, the probability of winding up with a favorable quantity of loan after n turns if you began without loan would certainly be $\frac{1}{2^n}{n\choose \left\lfloor\frac{n}{2}\right\rfloor}$ . Due to the fact that you start with one buck as opposed to no loan, this is countered to make sure that the probability of having loan after n turns is $\frac{1}{2^{n-1}}{n-1\choose \left\lfloor\frac{n-1}{2}\right\rfloor}$ for n ≥ 1.

This type trouble on arbitrary stroll is generally addressed with the Reflection Principle, with the stroll envisioned as a latticework course. Oddly, I can not locate an on-line reference to the remedy yet it is given up Feller's publication on probability concept, quantity 1.

Below, gauging the cash in devices of 0.5 bucks, the stroll, reeled in the $(x,y)$ aircraft, begins at (2,0), relocates by $(+1,\pm 1)$ at each action, and also the inquiry is what is the probability that over $n$ relocates the stroll is constantly $\geq 1$.

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