Calculate the probability of getting exactly 50 heads and 50 tails after flipping a fair coin 100 times.

(a) Calculate the probability of getting exactly 50 heads and 50 tails after flipping a fair coin 100 times (b) In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the expected number of girls each family has if you assume that having a girl or a boy is equally likely? Hint: if you use the definition of expected value, you may have to find the limit of an infinite suim (c) Suppose that a Bayesian spam filter is trained on a set of 10,000 spam messages and 5,000 messages that are not spam. The word "opportunity" appears in 800 spam messages and 100 messages that are not spam. 20% of the incoming email messages to the inbox are spam. Find the probability that a received message containing the word "opportunity" is spam