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Expected number of games in a series that terminates 
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#1
Oct3111, 06:13 AM

P: 9

The Question:
2 people A, B play a series of independent games. We have the following probabilities: P(A wins a game) = p P(B wins a game) = q = 1  p Both players begin with X units of money, and in each game the winner takes 1 unit from the other player. The series terminates when either A or B loses all their money. The assumption is that p > q. Derive the expected number of games in a series. Attempt at a solution: If: Z = number of games, then we are after E[Z]: So if the number of games is X and the series terminated, that means that either that A won all the games from the start or B won all the games from the start. e.g. For E[Z]=N p[itex]^{X}[/itex]q[itex]^{0}[/itex] or p[itex]^{0}[/itex]q[itex]^{X}[/itex] In order for the series to terminate, the number of wins either for A or B has to be X greater than the number of wins for the other player. e.g. If B wins 1 game, then A needs to win X+1 games in order for the series to terminate. I'm not sure whether my attempt actually helps in obtaining a solution, can anyone advise? Thanks. 


#2
Nov211, 12:24 PM

P: 9

I can confirm that I've solved this problem.



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