1. There is a 50-50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia. (a) If the queen has had three princes without the disease, what is the probability the queen is a carrier. (b) If there is a fourth prince, what is the probability that he will have hemophilia? 2. Suppose X1, X2, X3 are i.i.d. random variables, each uniformly distributed on {1, 2, 3}. Find the probability function for X1 + X2 + X3. That is, find P(X1 + X2 + X3 = k), for k = 3,..., 9 3. Every person in a group of 1000 people has a 1% chance of being infected by a virus. The process of being infected is independent from person to person. Using random variables, write expressions for the following probabilities and solve them with R. (a) The probability that exactly 10 people are infected. (b) That probability that at least 16 people are infected. (c) The probability that between 12 and 14 people are infected. (d) The probability that someone is infected.