Question

A student is taking a multiple-choice exam with 14 questions. Each question has four alternatives. If the student guesses on 10 of the questions, what is the probability she will guess at least 8 correct

Answer 1

0.042% probability she will guess at least 8 correct

Explanation:

For each question, there are only two possible outcomes. Either the student guesses it correctly, or he does not. The probability of a student guessing 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.

Each question has four alternatives.

One is correct, so
`p = (1)/(4) = 0.25`

If the student guesses on 10 of the questions, what is the probability she will guess at least 8 correct

This is
`P(X \geq 8)` when
`n = 10`. So

`P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)`

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

`P(X = 8) = C_(10,8).(0.25)^(8).(0.75)^(2) = 0.0004`

`P(X = 9) = C_(10,9).(0.25)^(9).(0.75)^(1) = 0.00002`

`P(X = 10) = C_(10,10).(0.25)^(10).(0.75)^(0) \cong 0`

`P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) = 0.0004 + 0.00002 = 0.00042`

0.042% probability she will guess at least 8 correct