Question

A student takes a multiple-choice test with 8 questions on it, each of which has 4 choices. The student randomly

guesses an answer to each question.

What is the probability that the student gets fewer than 2 questions correct?

Round to 3 decimal places

Blank 1:

Answer 1

A student takes a multiple-choice test with 8 questions on it, each of which has 4 choices. The student randomly guesses an answer to each question.

What is the probability that the student gets fewer than 2 questions correct?

Round to 3 decimal places.

0.367

Explanation:

Answer 2

0.367 = 36.7% probability that the student gets fewer than 2 questions correct

Explanation:

For each question, there are only two possible outcomes. Either he guesses the correct answer, or he does not. The probability of guessing the correct answer on a question is independent of any other question. This means that the binomial probability distribution is used 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.

8 questions on it

This means that
`n = 8`

4 choices.

One of them is correct, which means that
`\pi = (1)/(4) = 0.25`

What is the probability that the student gets fewer than 2 questions correct?

This is

`P(X < 2) = P(X = 0) + P(X = 1)`

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

`P(X = 0) = C_(8,0).(0.25)^(0).(0.75)^(8) = 0.1`

`P(X = 1) = C_(8,1).(0.25)^(1).(0.75)^(7) = 0.267`

`P(X < 2) = P(X = 0) + P(X = 1) = 0.1 + 0.267 = 0.367`

0.367 = 36.7% probability that the student gets fewer than 2 questions correct