A box contains 100 cards: 70 of them red, 30 Black. Cards are picked from the box, one by one, with replacement.

a) Suppose initially 6 cards are picked. We define X= number of black cards drawn. What is the probability that you would have at most one black card?

b) What is E(X)? What is Var(X)?

c) Suppose instead you go on drawing cards till you see a black card. What is the probability that you need to draw at least three cards for this to happen?

d) Suppose you are drawing cards till you get three black cards. What is the probability that you need exactly 7 draws for that to happen?

e) Suppose in part (a), you did the experiment without replacement. What would be your probability? Are the two probabilities "close"? if so, why?