An interesting problem illustrates why do we need statisticians in daily life!

﻿﻿1. A and B play a series of games with A winning each game with probability p. The overall winner is the first player to have won two more games than the other. Then what is the probability of A winning this game?

Let ${A_i=\left\{\begin{array}{ll}1& \mbox{at game i, A is 1 point ahead}\\ 0 & \mbox{at game i, A and B are even}\\ -1 & \mbox{at game i, A is 1 point behind} \end{array}\right.}$ Then

$\displaystyle P(S|A_i=1)=p+(1-p)P(S|A_{i+1}=0)$

$\displaystyle P(S|A_i=0)=p\times P(S|A_{i+1}=1)+(1-p)\times P(S|A_{i+1}=-1)$

$\displaystyle P(S|A_i=-1)=p\times P(S|A_{i+1}=0)$
Notice that ${P(S|A_i)=P(S|A_{i+1})}$, we can solve this and the result is

$\displaystyle P(S|A_i=0)=\frac{p^2}{(1-p)^2+p^2}$
We can see that this is just the starting point of the game when ${i=0}$. Then

$\displaystyle P(S)=P(S|A_0=0)=\frac{p^2}{(1-p)^2+p^2}=\frac p{\frac1p-2+2p}>p,\quad \mbox{if\ } p>\frac12$
What does this mean?

Well, in one word, if you have a better chance to win a game than your opponent, make a rule like this and you will have a better chance to win than just play one game!

1. JM 說道：

EE說他長大就會看的懂了