Some customers of a retail chain have a store credit card that earns them bonus gifts when they make purchases at the chain. Currently, 40 customers are shopping in a store in this chain. Of these,...

Some customers of a retail chain have a store credit card that earns them bonus gifts when they make purchases at the chain. Currently, 40 customers are shopping in a store in this chain. Of these, half already have a store credit card. If employees offer store credit cards to 6 of these, what is the probability that all of those chosen already have a card?

(a) Explain why it would not be appropriate to use a binomial model for the number who already have a card.

(b) A family of six is shopping in the store. Noting that
gives the number of ways of picking a subset of
 items out
, what is the probability that the six randomly selected shoppers are in this family?

(c) How many possible subsets of those already having a card might the employees select?

(d) Combine your answers to (b) and (c) to find the probability that all six of those offered credit already have a card.

(e) Use the ideas of conditional probability to find the probability in (d) by a different means that avoids the binomial coefficient. (Think of the sequence of picking the six consecutively from those with a store card.)