Question

44> 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 nCx gives the number of ways of picking a subset of x items out n, what is the probability that the six randomly selected shoppers are in this family?

(c) How many possible subsets of those already hav- ing 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 (Chapter 9) 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.)

Answer 1

a) The probability that all of those chosen already have a card is

6 selected from hose who already have card is selected without replacement while a binomial model has sampling with replacement. Hence Binomial model cannot be used.

b) The probability that the six randomly selected shoppers are in this family is

$\frac{\binom{6}{6}}{\binom{40}{6}}={\color{Blue}0.0000002605266}$

c) The number of possible subsets of those already having a card might the employees select is

$\binom{20}{6}={\color{Blue} 38,760}$

d) The required probabiity is obtained by the answers in part (b) and (c),

$\frac{\binom{6}{6}}{\binom{40}{6}}\binom{20}{6}=0.01$

e)

The number of ways of picking 6 from 20 is

$20\times 19\times 18\times ...\times 15$

The number of ways of picking 6 from 40 is

$40\times 39\times 38\times ...\times 35$

The required probability is

$\frac{20\times 19\times 18\times ...\times 15}{40\times 39\times 38\times ...\times 35}=0.01$