2. Prove that a binary tree with depth d has at most 2d leaves. 3. Lets say the following is the distribution of smokers in the population Smokes | Does not smoke | Total 0.18 0.49 0.14 0.51 0.32 1 That is, for example, the probability of a randomly selected person being female and a non-smoker is 0.37. First, identify the relevant probability space. Find the probability that the randomly selected person is: (1) Male given that the person smokes; Male Female Total 0.31 0.37 0.68 (2) Smokes given that the person is a male; (3) Female given that the person does not smoke. Are the events "being a female" and "does not smoke" independent? Can you derive the definition of independence of events using the conditional probability? 4. For two single experiments consider the probability spaces W = (S, P), W = (S2, P2) Performing the experiments independently is modeled by W = W x W = (S,P) (1) If both experiments are about dice and i [1: 12]. What are the probabilities that i eyes are up, that is, of the events A = {(a,b) | a+b=i} (2) Try this if you read upcoming slides: what is the expected value E(X) of the random variable X- a + b? Hint: be extremely lazy.