Question

A multiple-choice test has 27 questions. Each question has 5 possible answers, of which only one is correct. What is the probability that sheer guesswork will yield exactly 18 correct answers?

Answer 1

Answer:

Around 0% probability that sheer guesswork will yield exactly 18 correct answers

Step-by-step explanation:

For each question, there are only two possible outcomes. Either you guess the correct answer, or you do not. The probability of guessing the correct answer is 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.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

A multiple-choice test has 27 questions.

This means that $n = 27$

Each question has 5 possible answers, of which only one is correct.

This means that $p = \frac{1}{5} = 0.2$

What is the probability that sheer guesswork will yield exactly 18 correct answers?

This is P(X = 18).

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 18) = C_{27,18}.(0.2)^{18}.(0.8)^{9} = \cong 0$

Around 0% probability that sheer guesswork will yield exactly 18 correct answers