Question

At a particular supermarket, 80 percent of customers have the supermarket club card. Determine the probability that, in a random sample of 10 customers: a. Exactly 4 customers have the supermarket club card. B. At most 2 customers have the supermarket club card. C. At least 7 customers have the supermarket club card.

Answer 1

a) 0.55% probability that exactly 4 customers have the supermarket club card.

b) 0.01% probability that at most 2 customers have the supermarket club card.

c) 87.91% probability that at least 7 customers have the supermarket club card.

Explanation:

For each customer, there are only two possible outcomes. Either they have the supermarket club card, or they do not. The probability of a customer having the supermarket club card is independent of other customers. So we use the binoial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

80 percent of customers have the supermarket club card.

This means that
`p = 0.8`

10 customers.

This means that
`n = 10`

a. Exactly 4 customers have the supermarket club card.

This is P(X = 4).

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 4) = C_(10,8).(0.8)^(4).(0.2)^(6) = 0.0055`

0.55% probability that exactly 4 customers have the supermarket club card.

B. At most 2 customers have the supermarket club card.

`P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)`

`P(X = 0) = C_(10,0).(0.8)^(0).(0.2)^(10) \cong 0`

`P(X = 1) = C_(10,1).(0.8)^(1).(0.2)^(9) \cong 0`

`P(X = 2) = C_(10,2).(0.8)^(2).(0.2)^(8) = 0.0001`

`P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0 + 0 + 0.0001 = 0.0001`

0.01% probability that at most 2 customers have the supermarket club card.

C. At least 7 customers have the supermarket club card.

`P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)`

`P(X = 7) = C_(10,7).(0.8)^(7).(0.2)^(3) = 0.2013`

`P(X = 8) = C_(10,8).(0.8)^(8).(0.2)^(2) = 0.3020`

`P(X = 9) = C_(10,9).(0.8)^(9).(0.2)^(1) = 0.2684`

`P(X = 10) = C_(10,10).(0.8)^(10).(0.2)^(0) = 0.1074`

`P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.2013 + 0.3020 + 0.2684 + 0.1074 = 0.8791`

87.91% probability that at least 7 customers have the supermarket club card.