Question

A multiple choice test contains 20 questions with answer choices A, B, C and D. Only one answer choice to each question represent correct answer. Find the probability that a student will answer exactly 6 questions correct if he makes random guesses on all 20 questions.

Answer 1

Answer:

0.1686 = 16.86% probability that a student will answer exactly 6 questions correct if he makes random guesses on all 20 questions.

Step-by-step explanation:

For each question, there are only two possible outcomes. Either the students answer it correctly, or he/she does not. The probability of answering a question correctly 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.

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

And p is the probability of X happening.

20 questions:

This means that $n = 20$.

Answer choices are A, B, C and D:

Only one is correct, so $p = \frac{1}{4} = 0.25$

Find the probability that a student will answer exactly 6 questions correct if he makes random guesses on all 20 questions.

This is P(X = 6). So

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

$P(X = 6) = C_{20,6}.(0.25)^{6}.(0.75)^{14} = 0.1686$

0.1686 = 16.86% probability that a student will answer exactly 6 questions correct if he makes random guesses on all 20 questions.