A test consists of 10 multiple choice questions, each with five possible answers, one of which is correct. To pass the test a student must get 60% or better on the test. If a student randomly guesses, what is the probability that the student will pass the test?

Probability of an event is measure of its chance of occurrence. The probability that the student will pass the test is 0.0064 approx

How to find that a given condition can be modeled by binomial distribution?

Binomial distributions consists of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as

[tex]X \sim B(n,p)[/tex]

The probability that out of n trials, there'd be x successes is given by

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]

For the given condition, we can model the situation by binomial distribution where each question's answering is a Bernoulli trial, independent of other question's answer, and answering correct is tagged as success.

P(Answering correct) = probability of success = p = 1/5 (as each option is equally likely, and one correct option is there per 5 option available

Probability of failure = q = 1-p = 4/5

Number of trials = 10 (since 10 Multiple Choice Questions are there)

Thus, if we take X = number of questions answered correctly, then:

[tex]X \sim B(n=10, p = 1/5 = 0.2)[/tex]

For getting at least 60% marks, student needs to answer 60% of 10 questions correctly, which is 6 questions (at least).

P(Student will pass the test) = P(X ≥ 6)

[tex]P(X \geq 6) =P(X = 6) + P(X = 7) + P(X = 8) + P(X=9)+P(X = 10)\\\\P(X \geq 6) = \sum_{i=6}^{10} \: ^{10}C_i p^i q^{10-i} = \sum_{i=6}^{10} \: ^{10}C_i (0.2)^i (0.8)^{10-i} \approx 0.0064[/tex]

Thus, the probability that the student will pass the test is 0.0064 approx

Learn more about binomial distribution here:

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