1. Lisa shoots at a target. The probability of a hit in each shot is 1/2. Given a hit, the probability of a bull's-eye is p. She shoots until she misses the target. Let X be the total number of bull's-eyes Lisa has obtained when she has finished shooting; find its distribution.
2. Karin has an unfair coin; the probability of heads is p (0 p 1). She tosses the coin until she obtains heads. She then tosses a fair coin as many times as she tossed the unfair one. For every head she has obtained with the fair coin she finally throws a symmetric die. Determine the expected number and variance of the total number of dots Karin obtains by this procedure.