What's the probability that the last ball is a red?

Question

There are $8$ red ball, $5$ blue ball in a bag.

1. randomly pick one ball, remember the color, throw it away.
2. randomly pick another ball. If the color is different, put it back, return to step 1; else, throw it away and repeat this step.

Question: what is the probability that the last ball is red?

Answer 1

Claim. The last ball is red with probability ${1\over2}\,$, irrespective of the initial numbers of balls.

Proof. The game consists of several rounds. A round begins in state $1$ with $m\geq1$ red balls and $n\geq1$ blue balls. According to the rules we randomly pick balls until one of the following has been composed: $$\underbrace{\mathstrut B\,B\,\ldots\, B}_k\,R\qquad{\rm resp.}\qquad \underbrace{\mathstrut R\,R\,\ldots\, R}_k\,B\ .$$ We then discard the initial group of $k\geq1$ blue, resp., red balls. With probability $$p={n\over m+n}\cdot{n-1\over m+n-1}\cdot{n-2\over m+n-2}\cdots{1\over m+1}={n!\>m!\over(m+n)!}$$ all $n$ blue balls have been picked and thereafter discarded in this process. With the same probability $p$ all $m$ red balls have been picked and discarded. In these two cases the game has come to an end.

With probability $1-2p$ balls of both colors remain, and we return to state $1$ for the next round.

Since in each played round both colors have the same chance of winning the claim follows.

Answer 2

Let us consider $m$ red balls and $n$ blue balls. Let $u_{mn}(r)$ and $u_{mn}(b)$ be the probability that the sequence of balls being discarded should begin and end with a red ball and a blue ball respectively. We require $u_{mn} = u_{mn}(r) + u_{mn}(b)$.

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Consider $u_{mn}(r)$. The probability of choosing a red ball first is $\frac{m}{m+n}$. The probability of then getting a sequence ending in red is made up of two parts. Firstly, that the first and the last ball should be red is $u_{m-1,n}(r)$. Secondly, that the first ball should be blue namely $\frac{n}{m+n-1}$, multiplied by the probability of a sequence ending in red chosen from $m-1$ red and $n$ blue balls, remembering that the blue ball was replaced namely $u_{m-1,n}$. Hence for $m\geq 2, n\geq 1$, $$u_{mn}(r) = \frac{m}{m+n}[u_{m-1,n}(r) +\frac{n}{m+n-1}u_{m-1,n}]...(1)$$ $(1)$ also holds if $m=1, n\geq 1$. Similarly for $m\geq 1, n\geq 2$, $$u_{mn}(b) = \frac{n}{m+n}[u_{m,n-1}(b) + \frac{m}{m+n-1}u_{m,n-1}]...(2)$$. Since $u_{m,1}(b) =\frac{1}{m+1}$,$u_{m,0}(b) =0, u_{m,0} = 1$ equation $(2)$ holds if $m\geq 1, n=1$.

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The probability that the final ball should be blue is $1-u_{m,n}$. By interchanging the roles of the red and blue balls, it is easy to see that this is $u_{n,m}$. Hence, $$u_{m,n} + u_{n,m} = 1 (m\geq 1, n\geq 1)...(3)$$. Let $q_{m,n}(r) = \frac{(m+n-1)!}{m!n!} u_{m,n}(r) (m\geq 1, n\geq 1)$ with similar definitions for $q_{m,n}(b)$ and $q_{m,n}$. Writing $a$ in place of $m$ in $(1)$ and substituting for $u_{m,n}$, we have, $$(a+n)q_{a,n}(r)-(a+n-1)q_{a-1,n}(r) = nq_{a-1,n} (r\geq 1, n\geq 1)$$. Adding for $r=1,2...m$ and for $m\geq 1, n\geq 1$ we have $$(m+n)q_{m,n}(r)-nq_{0,n}(r) = n(q_{m-1,n}+q_{m-2,n}+...q_{0,n})...(4)$$ and for $m\geq 1, n\geq 1$, we have, $$(m+n)q_{m,n}(b)-mq_{m,0}(b) = m(q_{m,n-1}+q_{m,n-2}+...q_{m,0})...(5)$$. We also have $q_{0,n}(r)=0 (n\geq 1)$ and $q_{m,0}(b)=0 (m\geq 1)$. $(4)+(5)$ and $m\geq 1, n\geq 1$ gives $$(m+n)q_{m,n} =n(q_{m-1,n}+...q_{0,n}) +m(q_{m,n-1}+...q_{m,0})...(6)$$. If $n=1, m\geq 1$, we have $$(m+1)q_{m,1} = q_{m-1,1}+...q{0,1}+mq_{m,0} = q_{m-1,1}+...q_{1,1}+q_{0,1}+1$$. In terms of $u_{m,1} (m\geq 1)$, we have, $$(m+1)u_{m,1} = u_{m-1,1}+...u_{0,1}+1$$. Writing $m+1$ in place of $m$ and subtracting the two equations, we have, $$(m+2)u_{m+1,1}-(m+1)u_{m,1} = u_{m,1}$$ and so $u_{m+1,1} = u_{m,1} = u_{1,1} = \frac{1}{2}...(7)$.
Similarly by putting $m=1$ in $(6)$ we get $u_{1,n} (n\geq 1)$. Hence $u_{m,1} = u_{1,n} =\frac{1}{2}$ for $m\geq 1, n\geq 1$.

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We write equations by putting $m+1$ in place of $m$, $n+1$ in place of $n$ and combining $(m+1,n+1)$ in place of $(m,n)$. Subtracting $(6)$ and the sum of these three equations we have, $$(m+n)q_{m,n}-(m+n+1)(q_{m+1,n} + q_{m,n+1})-(m+n+2)q_{m+1,n+1} = -nq_{m,n} +(n+1)q_{m,n+1} + (m+1)q_{m+1,n} -mq_{m,n}$$. Reverting to $u_{m,n}$ for $m\geq 1, n\geq 1$, we have $$2(m+1)(n+1)u_{m,n} + (m+n+2)(m+n+1)u_{m+1,n+1} = (m+1)(m+2n+2)u_{m,n+1} + (2m+n+2)(n+1)u_{m+1,n}$$ $$\Rightarrow (m+n+2)(m+n+1)((u_{m+1,n+1}-u_{m,n+1})-(u_{m+1,n}-u_{m,n})) + m(m+1)(u_{m+1,n}-u_{m,n}) + n(n+1)(u_{m,n+1}-u_{m,n}) = 0$$ This implies that $u_{m,n} = c$(a constant)$(m\geq 1, n\geq 1)$ is a solution of the problem. It satisfies the boundary conditions of $(7)$ if $c=\frac{1}{2}$ hence $u_{m,n}$ is a solution of the problem. Hope it helps.