# A student takes a 20-question, true/false exam and guesses on each question. Find the probability of passing if the lowest passing grade is 15 correct out of 20. Would you consider this event likely to occur?Explain your answer.

Question

A student takes a 20-question, true/false exam and guesses on each question. Find the probability of passing if the lowest passing grade is 15 correct out of 20. Would you consider this event likely to occur?

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Step 1

Introduction to binomial distribution:

Consider a random experiment involving n independent trials, such that the outcome of each trial can be classified as either a “success” or a “failure”. The numerical value “1” is assigned to each success and “0” is assigned to each failure.

Moreover, the probability of getting a success in each trial, p, remains a constant for all the n trials. Denote the probability of failure as q. As success and failure are mutually exclusive, q = 1 – p.

Let the random variable X denote the number of successes obtained from the n trials. Thus, X can take any of the values 0,1,2,…,n.

Then, the probability distribution of X is a Binomial distribution with parameters (n, p) and the probability mass function (pmf) of X, that is, of a Binomial random variable, is given as:

Step 2

Obtain the probability that the student passes in the exam:

It is given that a student’s takes a 20 questions test. That is, the size of the sample of the questions is n = 20. It is known that each question will not depend on the other question, imp-lying independence among each other. As a result, the 20 questions may be considered as 20 independent trials.

Consider the event of answering the question correctly as a “success”. It is said that, all the 20 questions in the exam are true/false questions and the student answers by guessing. Thus, the probability that the guess of the student is correct or the student answers correctly, that is, the probability of success in each trial is p = 1/2. Then q = 1 – (1/2) = 1/2.

Consider X as the number of questions answered correctly among the 20 questions. Then, X has a Binomial distribution with parameters (n = 20, p = 1/2) with pdf as given below:

P(x) = 20Cx*(1/2)x * (1/2)20–x ; x = 0, 1 , 2 , ...,20.

Moreover, it is given that the lowest passing grade is 15 correct out of 20 questions.

The requirement is to find the probability of passing. Here, the lowest passing grade is 15 correct out of 20 questions. That is, if the student answers 15 or more questions (X ≥ 15) then, the student passes.

The probability of passing is obtained as 0.0207 from the calculation given below:

Step 3

Check whether the event is likely to occur:

The probability of passing is 0.0207. This probability value is very low.

There...

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