1. Show that if Pr(A|B) = Pr(A|Bc) then Pr(A|B) = Pr(A|B). This means that since if Pr(A|B) = Pr(A|B) is true only if A and B are independent we also have Pr(AB) = Pr(A|B) is true only if A and B are independent. 2. Show that if Pr(A|B) = Pr(A) then Pr(A|B) = Pr(AC). This means that since if Pr(A|B) = Pr(A) is true only if A and B are independent we also have Pr(A|B) = Pr(A) is only true if A and B are independent. 5. Suppose that 1/3 of new businesses fail in one year. Of those remaining after one year, 1/4 fail in their second year. (a) What is the probability that a new business will survive for two years? (b) What is the probability that a new business will fail in its second year? Hint: To fail in the second means you have to have survived the first. 6. Suppose events A and B are not independent. For each of the following, say if the statement is true or false. (a) Pr(Ac|B) # Pr(A) (b) Pr(A/B) # Pr(A) (c) Pr(A|B) # Pr(Ac) (d) Pr(A and B) = Pr(A)Pr(B) 7. A box contains 4 coins coin 1 has both sides tails coin 2 has both sides heads coin 3 has both sides heads coin 4 is a regular coin (1 side head, other side tails) (a) If we randomly choose one coin from the box and flip, what is the probability we get heads? (b) Suppose we randomly choose a coin, flip, and get heads. What is the probability that the coin that was chosen is the regular (1 side head, other side tails) coin?