A student takes a multiple-choice test with 10 questions and guesses randomly at each answer. Each question has 5 choices with only one correct choice. (a) (2 points) Find the probability that he/she gets exactly 3 correct answers.

The probability that he/she gets exactly 3 correct answers is 0.2013

Step-by-step explanation:

In this question, we are tasked with calculating the probability that out of 10 questions, a student who is a random guesser, gets exactly 3 correct questions given that the test is a multiple-choice test with 5 options each.

The total number of questions n = 10

Now let the probability of answering one question correctly be p, since there is exactly one correct answer out of 5 options, this probability is equal to 1/5

Let the probability of picking an incorrect option be q, since there are 4 incorrect options, the probability of selecting an incorrect option is 4/5

Now, we compute the probability of getting exactly 3 correct options using the Bernoulli test as follows;

Probability that he/she gets exactly 3 correct answers P(X=3) =nCx × p^x × q^(n-x)

→ 10C3 × (1/5)^3 × (4/5)^7 = 0.2013

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