James is a DJ at local radio station. One day, James offers a cash prize to the first caller who correctly answers a rather difficult trivia question. James’ assistant screens and records all incoming calls, and lets James know when the first call with the correct answer has been received; only then does James air the recording of the caller responding correctly. The probability that a randomly selected caller will correctly answer the trivia question is 0.1.

What is the probability that the third caller is the first to correctly answers the question?

0.01

0.0729

0.081

0.09

0.9

Using the inverse binomial distribution, it is found that there is a 0.081 = 8.1% probability that the third caller is the first to correctly answers the question.

The number of callers that correctly answer the question is a binomial variable, however, we want the probability of the third caller being the first to answer the question correctly, hence it is a negative binomial variable.

Inverse binomial:

It is the number of trials until q successes of a binomial variable, with p probability of success.

The probability mass function is:

[tex]P(X = x) = C_{x+q-1,q-1}(1 - p)^xp^q[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this question:

Number of trials until 1 success, hence [tex]q = 1[/tex].

0.1 probability of a success on each trial, hence [tex]p = 0.1[/tex].

The probability that the third caller is the first to correctly answers the question is P(X = 3), hence:

[tex]P(X = x) = C_{x+q-1,q-1}(1 - p)^xp^q[/tex]

[tex]P(X = 3) = C_{3,0}(0.9)^2(0.1)^1 = 0.081[/tex]

0.081 = 8.1% probability that the third caller is the first to correctly answers the question.

To learn more about the inverse binomial distribution, you can take a look at https://brainly.com/question/14581720