Please help 1 Suppose you play a certain game and win with p

Please help

1. Suppose you play a certain game and win with probability p. You plan to play 5 games, but if you win the fifth game, then you will keep on playing until you lose. Let the random variable X be the number of games you play. (a) Find the probability mass function for X. (b) Find the probability that you will play exactly 8 games given that you win on the fifth game. (c) Find the expected number of games that you play.

Solution

The result of the first 4 games doesn\'t matter; if you win the 5th game, you\'ll keep going until you lose.

The chances of playing the 6th game is p, the 7th is p^2, etc.

So the player will play 5+ p + p^2 + p^3 ... games,
or 4 + (1+ p + p^2 + p^3 .......)
or 4 + sum (0 to infinity) p^n
or 4 + 1/(1-p)
or 5 + p/(1-p)
The expected number of games that will be played  = 4 + 1/(1-p)