Question

A student is given 20 multiple choice questions. Each multiple choice question has 4 options & only one correct answer. The student answers all the questions in a random manner.

a) What is the probability the student scores 80% or above?

b) What is the probability the student scores 50% or above?

Answer 1

Solution

Back-up Theory

Implication of random selection

One item is picked (selected) at random from n items => each of n items has equal chance of being

selected and hence P(a particular item is selected) = 1/n .................................................................................................... (1)

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and

p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ………….........................................................................………..(2)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST].............................………….(2a)

Now, to work out the solution,

Let X = number of questions answered correctly by the student.

‘Each multiple choice question has 4 options & only one correct answer. The student answers all the questions in a random manner.’ implies vide (1), P(an answer is correct) = ¼ = 0.25.

Also given that there are 20 questions to be answered.

Then, X ~ B(20, 0.25)

Part (a)

Since 80% of 20 is 16, the probability the student scores 80% or above

= P(X ≥ 16)

= 0.00000039 [vide (2) and (2a)] Answer 1

Part (b)

Since 50% of 20 is 10, the probability the student scores 50% or above

= P(X ≥ 10)

= 0.0139 [vide (2) and (2a)] Answer 2

DONE