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

Also, probability of not getting a 6

Now, there are three cases from which the expected value of the amount which he wins can be calculated:

(i) First case is that, if he gets a six on his first through then the required probability will be

∴ Amount received by him = Rs. 1

(ii) Secondly, if he gets six on his second throw then the probability

∴ Amount received by him = - Rs. 1 + Rs. 1

= 0

(iii) Lastly, if he does not get six in first two throws and gets six in his third throw then the probability

∴ Amount received by him = - Rs. 1 – Rs. 1 + Rs. 1

= - 1

Hence, expected value that he can win

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