Question

Topic: (Renewal Models and Diffusion Processes)

Tom plays Jerry in a chess tournament, which is won by the first player who wins two consecutive games. Probability of Tom winning each game is p=0.6 and there are no drawn games. Let W be the number of games("time") required before the tournament is won. By writing down (in terms of p), P[W>1], P[W>2], P[W>3], P[W>4], P[W>5], and P[W>2m], P[W>2m+1] where m is a positive integer, calculate the Expected Duration (in number of games) of the tournament to 2 decimal places).

Pls do not cut n paste from similar question as the solution given didn't answer the acutal question ask... Asking to find Expected Duration (in number of games).

Pls explain with workings. Thxs

Answer 1

Given the probability of Tom winning each game is .So the probability of Jerry winning each game is since no game is drawn.

The event that the game ends in the game can be

1) Tom wins the -the game, won the game. games they win alternately.

The probability of win is

2) Jerry wins the -the game, won the game. games they win alternately.

The probability of win is

The events in case (1) and (2) are disjoint. The probability that the game ends in -the game is the sum of the above probabilities.

That is is an even number and greater than 0. The probability that the game will end in odd number of games is 0.

We have to normalize the above probability by dividing with

Since the probabilities should add to 1.

The expected value of the number of games for one of them to wins is (use expectation formula of geometric distribution)

Very hard work. Kindly upvote.