Question

A test contains twenty true false questions. A student either knows the answer to a question, or guesses at random if he does not know the answer. On any question, the student knows the answer with probability .7. What is the probability that he gets exactly 2 of the twenty questions wrong, if his performance on different questions is independent

Answer 1

2.78% probability that he gets exactly 2 of the twenty questions wrong

Explanation:

For each question, there are only two possible outcomes. Either he knows the answer, or he does not. The probability of him knowing the 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.

On any question, the student knows the answer with probability .7.

This means that
`p = 0.7`

What is the probability that he gets exactly 2 of the twenty questions wrong, if his performance on different questions is independent

2 of 20 wrong, 20-2 = 18 correctly. So this is P(X = 18).

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

`P(X = 18) = C_(20,18).(0.7)^(18).(0.3)^(2) = 0.0278`

2.78% probability that he gets exactly 2 of the twenty questions wrong