When John and Carl play a game,John wins,on average,2 out of every 5 games played.Today they agree to play until Carl wins one game or until three games have been played,whichever comes first.At the end of each game,the loser pays the winner $1.What is Carl's expected winnings?
A)0.264
B)0.312
C)0.504
D)0.936

Question

Quizplus · Accepted Answer

There are three possible outcomes:
1. John wins in the first game
2. John wins in the second game
3. Carl wins in the third game

The first two outcomes have a probability of (2/5) * (3/5) = 6/25 each.
The third outcome has a probability of (3/5) * (2/5) * (3/5) = 18/125.

In the first outcome, Carl loses $1.
In the second outcome, Carl loses $2.
In the third outcome, Carl wins $1.

Therefore, Carl's expected winnings are:

(6/25)*(-1) + (6/25)*(-2) + (18/125)*(1) = -6/25 + (-12/25) + 18/125 = 39/125 = 0.312

When John and Carl Play a Game,John Wins,on Average,2 Out