Question

A multiple choice exam has ten questions. Each question has five possible answers, of which one is correct. Suppose that a student did not study for the exam and, as a result, they guess on every question so that the probability of answering any question correctly is 0.20. 19. What is the probability that the student answers exactly 4 questions correctly

Answer 1

8.81% probability that the student answers exactly 4 questions correctly

Explanation:

For each question, there are only two possible outcomes. Either he answers it correctly, or he does not. The probability of answering a question correctly 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.

A multiple choice exam has ten questions.

This means that
`n = 10`

The probability of answering any question correctly is 0.20.

This means that
`p = 0.2`

What is the probability that the student answers exactly 4 questions correctly

This is P(X = 4).

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

`P(X = 4) = C_(10,4).(0.2)^(4).(0.8)^(6) = 0.0881`

8.81% probability that the student answers exactly 4 questions correctly