Two integers are drawn independently at random from 1, 2,..., 100 (repetition is allowed, for instance, both numbers could be 1). Consider the following events. E: both numbers are even E2: the first number is 100 E5: both numbers are divisible by 11 For any pair of these events decide whether the two events are independent. Problem 3 (conditional probability) (a) For the events E1,..., Es from Problem 3, compute the following conditional probabilities. P(E1 E2), P(E1 E3), P(E4 E3), P(E1 E5), P(E4 E2), P(E2E4). (b) A dice is rolled 4 times. What is the probability to roll a total of at least 22 under the condition that at least two of the rolls are a 6? (c) A family has two children. What is the probability that both children are girls under the condition that the first child is a girl? What is the probability that both children are girls under the condition at least one of the children is a girl? (Here we assume that the probability for a child to be a girl is 1.)