1. Two riflemen A and B are shooting at a target. Independently of who is shooting, the probability that the shot results in a hit is 0.5. Each shot is independent from others, and the riflemen shoot at the target one by one in the order A, B, A, B, ... . What is the probability that A hits the target before B?
2. Conditional Probability. Problem 2.73, page 88 of ALG.
(a) Find P(A|B) if A ∩ B = ∅; if A ⊂ B; if A ⊃ B.
(b) Show that if P(A|B) P(A), then P(B|A) P(B).
3. The Number of Children. A family has j children with probability pj, where p1 = .1, p2 = .25, p3 = .35, p4 = .3. A child from this family is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family has (a) only 1 child; (b) 4 children.
4. Pairwise independence and overall independence. Alice, Bob and Claire each throw a fair die once. Show that the events A,B and C where A: “Alice and Bob roll the same face”, B: “Alice and Claire roll the same face” and C: “Bob and Claire roll the same face” are pairwise independent but not independent.
5. A binary Z-channel is show in the figure. Assume the input is “0” with probability p and “1” with probability 1 − p.
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(a) What can you say about the input bit if “1” is received?
(b) Find the the probability that the input was “1” given that the output is “0”.
