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!

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1 則回應給 An interesting problem illustrates why do we need statisticians in daily life!

  1. JM 說:

    EE說他長大就會看的懂了

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