Question

In a game a man wins 1 rupee for a six and loses one for any other number when a fair dice is thrown.man decided to throw dice thrice and quit as soon he wins.Find expected value he wins.

Won't we consider all loses in this ?

Answer 1

We will have to consider the case when he loses 3 times.

Let us understand the situation here--
The man throws a die.The first time,if he gets a six,he quits or else he throws the die the second time,if he gets a six,he quits or else he throws the die the third time.Now he quits in either case,wheather he gets a six or not.

Lets X be a random variable that takes values according to his gain.So X=-3,-1,0,1.(if the man gets a six in the first throw he has won Re 1,if he gets a number other than 6 and a six in the second,he lost Re 1 and then got Re 1,so he gained Rs 0.Use this notation to find the other values)

Now the probabilities of different values of X are -
-3 = 5/6 x 5/6 x 5/6
-1=5/6 x 5/6 x 1/6
0=1/6 x 1/6
1=1/6
(This is derived from the fact that if he gets a six then the probability here will be 1/6 else 5/6)

Now expected value of X=Mean of X=summation(XP(X))
125/216 x -3 + 25/216 x -1 + 1/36 x 0 + 1/6 x 1=-91/54