**Discrete Math

1.

In a large lecture hall of 50 students, an evil math professor makes a bet against the students. He bets that at least two students share a birthday, claiming that the odds of this happening are 49 / 365 (13.4%) since each student is in the same situation of getting 49 chances to match a birthday out of the 365 total birthdays possible. To be sporting, he even gives 2:1 odds, offering $2 to every single student ($100 total) if he is wrong, but will collect only $1 from each student if he is right.

Part 1 - Assuming that having one birthday is just as likely as any other (and that the students don't lie too!) what is the actual probability that two students in this class share a birthday?

Part 2 - What is the expected value of the bet to the professor? ?

Part 3 - What is the expected value to a student?

Please explain the details, show all work, and round to the nearest penny!

2.

Let A be a non-empty set. Prove or disprove: the number of subsets of A whose size is even is equal to the number of subsets of A whose size is odd. To be clear, here's an example... A = {1, 2}. The subsets of A are {}, {1}, {2}, and {1, 2}. Since there are two subsets of odd size ({1} and {2}) and two subsets of even size ({} and {1, 2}) the claim holds for this particular example