HW7 (1) Philippus flips a fair coin 100 times. Let the outcome be the number of heads that he sees. (a) What is the sample space? (b) What is Pr[0]? Philippus now flip his fair coin n times. He is interested in the event \"there are (strictly) more heads than tails.\" What's the probability of this event for the following values of n? (c) n = 2 (d) n = 3 (2) A bitstring x {0, 1}5 is stored in vulnerable memory, subject to corruption - for example, on a spacecraft. An -ray strikes the memory and resets one bit to a random value (both the new value and which bit is affected are chosen uniformly at random). A second -ray strikes the memory and resets one bit (again chosen uniformly at random). What is the probability that the resulting bitstring is identical to x? (3) Argue briefly that the following properties hold. (a) For any outcome s S, we have Pr[s] 1. (b) For any event A S, we have Pr[A] = 1 Pr[A]. (Recall that A = S A.) (c) For any events A, B S, we have Pr[A B] = Pr[A] + Pr[B] Pr[A B]. (4) We hash items into a 10-slot hash table using a hash function h that uniformly assigns elements to {1, 2, . . . , 10}. Compute the probability of the following events if we hash 3 elements into the 10-slot table. (a) no collisions occur (b) all 3 elements have the same has value (5) We flip a fair coin 6 times. Which of these events are independent or dependent? Justify your answers. (a) \"The number of heads is even\" and \"the number of heads is divisible by 3\" (b) \"The number of heads is even\" and \"the number of heads is divisible by 4\