**Exercise 3.28.** There are five closed boxes on a table. Three of the boxes have nice prizes inside. The other two do not. You open boxes one at a time until you find a prize. Let $ X $ be the number of boxes you open.(a) Find the probability mass function of $ X $.(b) Find $ E[X] $.(c) Find $ \text{Var}(X) $.(d) Suppose the prize inside each of the three good boxes is $100, but each empty box you open costs you $100. What is your expected gain or loss in this game?**Hint.** Express the gain or loss as a function of $ X $.

a) Let X represent the number of boxes you open until you find a prize.P(X=1) is the probability that the first box you open contains a prize. Since there are 3 prizes among 5 boxes, P(X=1)=3 / 5.P(X=2) is the probability that the first box is empty (2 / 5), and the second box contains a prize. Now, there are 3 prizes left and 4 boxes, so P(X=2)=(2 / 5) *(3 / 4).P(X=3) follows a similar pattern. The first two boxes are empty, and the third contains a prize. Thus, There is no or because you stop once you find a prize.So the probability mass function (PMF) of X is:b) To find the expected value E[X], you multiply each value of X by its probability and sum those products:c) To find the variance, we first need the expected value of X squared, Math input errorMath input errord) The gain or loss can be represented as a function of X :If X=1, you win $ 100.If X=2, you've paid $ 100 for the first box and won $ 100 for the second box, so the net gain is $ 0.If X=3, you've paid $ 200 in total for the first two boxes and won $ 100 for the third box, so the net loss is -$ 100.Now, to find the expected gain or loss, you multiply each outcome by its probability and sum them:Expected Gain/Loss =( 100 * P(X=1))+(0 * P(X=2))-( 100 * P(X=3))Expected Gain/Loss =( 100 * 3 / 5)+( 0 * 3 / 10)-( 100 * 1 / 10)Expected Gain/Loss =$ 60-$ 10Expected Gain/Loss =$ 50So the expected gain is $ 50.a) b) c) d) So the expected gain is $ 50.