Question

Richard has been given a 12-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended the class recently, he doesn't know any of the answers. The success occurs if Richard answers a question correctly and the failure occurs if Richard is unable to answer a question correctly. Assuming that Richard guesses on all 12 questions, find the probability that he will answer no more than 3 questions correctly. Round your answer to the nearest thousandth.

Answer 1

0.649 = 64.9% probability that he will answer no more than 3 questions correctly.

Explanation:

For each question, there are only two possible outcomes. Either Richard guesses the correct answer, or he does not. The probability of Richard guessing the correct answer in a question is independent of any 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.

12 questions.

This means that
`n = 12`

Each one with four options, one of which is correct.

This means that
`p = \frac{1}[4} = 0.25`

Assuming that Richard guesses on all 12 questions, find the probability that he will answer no more than 3 questions correctly.

`P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)`

In which

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

`P(X = 0) = C_(12,0).(0.25)^(0).(0.75)^(12) = 0.032`

`P(X = 1) = C_(12,1).(0.25)^(1).(0.75)^(11) = 0.127`

`P(X = 2) = C_(12,2).(0.25)^(2).(0.75)^(10) = 0.232`

`P(X = 3) = C_(12,3).(0.25)^(3).(0.75)^(9) = 0.258`

`P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.032 + 0.127 + 0.232 + 0.258 = 0.649`

0.649 = 64.9% probability that he will answer no more than 3 questions correctly.