SOLVE WITH EXPLANATION

Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility that the first k terms are all distinct is n1 712 71 kl P(n,k)= n ' n """ #5 as we saw in class. For this problem, you will want to write a (very short) program to compute this number, but you do not need to submit it this time. (For both parts, you can submit a numerical approximation to the answer.) a) Suppose that n = 365. What is the smallest k for which P (365, k) is less than 1/2? (This is the solution to the \"birthday problem\": if k people are in a room, there is a greater than 50% chance that two of them have the same birthday. b) Suppose that n = 1000. What is the smallest k for which P(1000, k) is less than 1/100