Question

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.

Answer 1

Hint: In this question, we will find all the cases in which the man can win or lose the game and then we will find the probability and the amount he will receive.

Complete step-by-step answer:
It is given in the question that the man will get Re. 1 if he will get a six and will lose Re. 1 if he will not get six.
So, probability of getting a six = $\dfrac{1}{6}$
Probability of not getting a six = 1-$\dfrac{1}{6}$=$\dfrac{5}{6}$
Now, we have three cases according to the question as follows:
E. If the man gets a 6 in the first throw, he will win, then the required probability = $\dfrac{1}{6}$
In this case he will receive Re. 1
F. If the man would not get a 6 in the first throw and gets a six in the second throw, then the probability would be $\dfrac{5}{6} \times \dfrac{1}{6}$=$\dfrac{5}{{36}}$
In this case, first he will lose Re. 1 and then he will win Re. 1 So, -1+1=0
He will not receive any amount.
G. If the man would not get a 6 in the first two throws and gets a six in the third throw, then the probability would be $\dfrac{5}{6} \times \dfrac{5}{6} \times \dfrac{1}{6}$=$\dfrac{{25}}{{216}}$
In this case, he will receive amount -1+(-1)+1=-1
He will lose Re. 1

Now, we have to find the expected value the man will win or lose
Let us know What is the Expected value?
The expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values.
So, Expected Value he can win = $\dfrac{1}{6}(1) + (\dfrac{5}{6} \times \dfrac{1}{6})(0) + [\dfrac{5}{6} \times \dfrac{5}{6} \times \dfrac{1}{6}]( - 1)$
                    = $\dfrac{1}{6} - \dfrac{{25}}{{216}}$
                    = $\dfrac{{36 - 25}}{{216}}$
                    = $\dfrac{{11}}{{216}}$= 0.05

Note: Multiply the probability of all the cases with the amount calculated in all the three cases and add them to get the final result.