Stats Help Please!!

Question

In a game, there are 5 boxes when only two of them have a ticket and others are empty. You are given a chance to select two boxes randomly and only if both of your selections have the tickets you win the reward.

a) What is the probability of wining?

b) After you have chosen two, the host opens one of the remaining 3 boxes that he knows is empty and you are given a chance to swap one of your boxes. What is the probability of winning if you do so?

c) What is the probability of winning if you swap both of your boxes?

Answer 1

This is a revision of the Monty Hall problem

a)As there are 5 boxes, there are C(5,2) = 10 pairs of boxes, so the probability that you have chosen the correct pair is 1/10

b) As he can always pick an empty box, this does not change the probability that your pair is the correct pair, so the probability remains 1/10. Then, the probability of one of the other pairs winning is 9/10. However, since since we have eliminated one of the boxes, we have eliminated 4 pairs (the box eliminated paired with each of the other boxes). Thus, there are 5 boxes remaining, so each has 1/5 * 9/10 = 9/50 chance of winning.

c) Swapping two boxes gives you the same chance as swapping one box. You are choosing one of the 5 pairs that you did not originally select, which has probability 9/50.