Scratch off. A retail company offers a "scratch off" promotion. Upon entering the store, you are given a card. When you pay, you may scratch off the coating. The company advertises that half the cards are winners and have immediate cash-back savings of $5 (the others offer $1 off any future purchase of coffee in the cafe). You aren't sure the percentage is really 50% winners.

a) The first time you shop there, you get the coffee coupon. You try again and again get the coffee coupon. Do two failures in a row convince you that the true fraction of winners isn't 50%? Explain.

b) You try a third time. You get coffee again! What's the probability of not getting a cash savings three times in a row if half the cards really do offer cash savings?

c) Would three losses in a row convince you that the store is cheating?

d) How many times in a row would you have to get the coffee coupon instead of cash savings to be pretty sure that the company isn't living up to its advertised percentage of winners? Justify your answer by calculating a probability and explaining what it means.