Question

A multiple choice exam has ten questions. Each question has five possible answers, of which one is correct. Suppose that a student did not study for the exam and, as a result, they guess on every question so that the probability of answering any question correctly is 0.20. 19. What is the probability that the student answers exactly 4 questions correctly

Answer 1

Answer:

8.81% probability that the student answers exactly 4 questions correctly

Step-by-step explanation:

For each question, there are only two possible outcomes. Either he answers it correctly, or he does not. The probability of answering 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.

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

And p is the probability of X happening.

A multiple choice exam has ten questions.

This means that $n = 10$

The probability of answering any question correctly is 0.20.

This means that $p = 0.2$

What is the probability that the student answers exactly 4 questions correctly

This is P(X = 4).

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

$P(X = 4) = C_{10,4}.(0.2)^{4}.(0.8)^{6} = 0.0881$

8.81% probability that the student answers exactly 4 questions correctly