[Solved] EE 131A Probability Discussion Set 2

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. 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.
   • Find P(A|B) if A B = ; if A B; if A 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 p_j, where p₁ = .1, p₂ = .25, p₃ = .35, p₄ = . 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.

1

• What can you say about the input bit if 1 is received?
• Find the the probability that the input was 1 given that the output is 0.