Question

On a multiple-choice test. Abby randomly guesses on all seven questions. Each question

has four choices. Find the probability to the nearest thousandth, that Abby gets exactly
three questions correct.

Answer 1

0.173 probability that she gets exactly three questions correct.

Explanation:

For each question, there are only two possible outcomes. Either she guesses the correct answer, or she does not. The probability of guessing the correct answer for a question is independent of other questions. 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.

Seven questions:

This means that
`n = 7`

Each question has four choices.

Abby guesses, which means that
`p = \frac{1}[4} = 0.25`

Find the probability to the nearest thousandth, that Abby gets exactly three questions correct.

This is P(X = 3).

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

`P(X = 3) = C_(7,3).(0.25)^(3).(0.75)^(4) = 0.173`