Question

a child wins a video game with probability p, and the outcome on any one game is independent of that on any other game. he decides to play four games. however, if he loses the fourth game, he will continue to play until he wins a game. a find the expected number of games that the child will play. b find the expected number of games that the child will win

Answer 1

To compute the expected number of games played and won when the probability of winning each game is p and the child continues to play after a loss until a win, we consider two scenarios contributing to the total expectation. The expected number of games played is 4 + (1 - p)(1/p), and the expected number of wins is 4p + (1 - p).

Step-by-step explanation:

Considering a game where the probability of winning is p and losing is 1 - p, if a child plays four games but continues to play after the fourth game until he wins, we need to calculate the expected number of games played and the expected number of games won.

To find the expected number of games played, we consider two scenarios: 1) the child wins the fourth game, and 2) the child loses the fourth game and continues to play until he wins. The expected number after the fourth game is given by 1/p since it's a geometric random variable with the probability of success p. The overall expected number of games played is 4 + (1 - p)(1/p).

To find the expected number of games won, we need to establish that the expected wins in the first four games is 4p, and if the child loses the fourth game, they will continue playing until they win, adding just one expected win. Therefore, the overall expected number of games won is 4p + (1 - p).