Question

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses

Answer 1

The man may get number greater than 4 in the first throw and then he quits the game. He may get a number less than equatl to 4 in the first throw and in the second throw he may get the number greater than 4 and quits the game.
In the first two throws he gets a number less than equal to 4 and in the third throw he may get a number greater than 6. He may not get number greater than 4 in any one of three throws.
Let X be the amount he wins/looses.

Then, X can take values -3, 3, 4, 5 such that
P (X = 5) = P(Getting number greater than 4 in first throw) = 1/3
​P (X = 4) = P(Getting number less than equal to 4 in the first throw and number greater than 4 in second throw) `= 4/6×2/6=2/9`

​P (X = 3) = P(Getting number less than equal to 4 in the first two throws and number greater than 4 in third throw) `= 4/6×4/6×2/6=4/27`

​P (X = -3) = P(Getting number less than equal to 4 in all three throws) `= 4/6×4/6×4/6=8/27`

X	5	4	3	-3
P(X)	`1/3`	`2/9`	`4/27`	`8/27`

`E (X) = (5×1/3)+4 (2/9)+3(4/27)−3 (8/27)`

`=1/27(45+24+12−24)`

 

`=57/27`