Question

According to a Rasmussen survey, 23% of adults responded that vanilla was their favorite ice cream flavor. A random sample of three adults was selected. Determine the probability that vanilla ice cream is the favorite flavor for all three adults.

Answer 1

1.22% probability that vanilla ice cream is the favorite flavor for all three adults.

Explanation:

For each adult, there are only two possible outcomes. Either vanilla is their favorite ice cream flavor, or it is not. The probability of an adult having vanilla as their favorite ice cream flavor is independent of other adults. So we use the binomial 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.

23% of adults responded that vanilla was their favorite ice cream flavor.

This means that
`p = 0.23`

A random sample of three adults was selected.

This means that
`n = 3`

Determine the probability that vanilla ice cream is the favorite flavor for all three adults.

This is P(X = 3).

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

`P(X = 3) = C_(3,3).(0.23)^(3).(0.77)^(0) = 0.0122`

1.22% probability that vanilla ice cream is the favorite flavor for all three adults.