In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.
In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.

For the situation given in the equation, we have

Probability of getting a six in a throw of a die =1 / 6

Also, probability of not getting a 6=5 / 6

Following that, there are three scenarios from which it is possible to compute the expected value of the amount that he wins:

As an example, consider the following scenario: He receives a six on his first try, and so meets the required probability be 1 / 6

\therefore Amount received by him = Rs. 1

(ii) Secondly, if he gets six on his second throw then the probability =(5 / 6 \times 1 / 6)

=5 / 36

\therefore Amount received by him = – Rs. 1 + Rs. 1

=0

(iii) Finally, if he does not get a six in his first two throws but does get a six in his third throw, the probability of getting a six is increased. =5 / 6 \times 5 / 6 \times 1 / 6

=25 / 216

\therefore Amount received by him = – Rs. 1 – Rs. 1 + Rs. 1

=-1

Hence, expected value that he can \operatorname{win}=1 / 6-25 / 216

=(36-25) / 216

=11 / 216