Question

According to the American Academy of Cosmetic Dentistry, 75% of adults believe that an unattractive smile hurts career success. Suppose that 25 adults are randomly selected. What is the probability that 15 or more of them would agree with the claim?

Answer 1

0.9703 = 97.03% probability that 15 or more of them would agree with the claim.

Explanation:

For each adult, there are only two possible outcomes. Either they agree with the claim, or they do not. The probability of an adult agreeing with the claim is independent of any other adult, which means that the binomial probability distribution is used 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.

75% of adults believe that an unattractive smile hurts career success.

This means that
`p = 0.75`

Suppose that 25 adults are randomly selected.

This means that
`n = 25`

What is the probability that 15 or more of them would agree with the claim?

This is:

`P(X \geq 15) = 1 - P(X < 15)`

In which:

`P(X < 15) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 13) + P(X = 14)`

14 is below the mean, so we start below and go until the probability is 0. Then

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

`P(X = 14) = C_(25,14).(0.75)^(14).(0.25)^(11) = 0.0189`

`P(X = 13) = C_(25,13).(0.75)^(13).(0.25)^(12) = 0.0074`

`P(X = 12) = C_(25,12).(0.75)^(12).(0.25)^(13) = 0.0025`

`P(X = 11) = C_(25,11).(0.75)^(11).(0.25)^(14) = 0.0007`

`P(X = 10) = C_(25,10).(0.75)^(10).(0.25)^(15) = 0.0002`

`P(X = 9) = C_(25,9).(0.75)^(9).(0.25)^(16) \approx 0`

Then

`P(X < 15) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) = 0.0002 + 0.0007 + 0.0025 + 0.0074 + 0.0189 = 0.0297`

And

`P(X \geq 15) = 1 - P(X < 15) = 1 - 0.0297 = 0.9703`

0.9703 = 97.03% probability that 15 or more of them would agree with the claim.