An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

Question

An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

Philoid · Accepted Answer

Here, there are two types of questions, True/False or Multiple Choice Questions (T/F or MCQ), and each of them are divided into Easy and Difficult type, as shown below in the tree diagram.

So, in all, there are, 500 T/F questions and 900 MCQs.

Also, there are 800 Easy questions and 600 Difficult questions.

⇒ The sample space of this experiment has 500 + 900 = 1400 outcomes.

Now,

Let E be the event of ‘getting an Easy question’ and F be the event of ‘getting an MCQ’.

and ……….(i)

Now, E ∩ F is the event of getting an MCQ which is Easy.

Clearly, from the diagram, we know that there are 500 MCQs that are easy.

So, ……….(ii)

Now, we know that

By definition of conditional probability,

[Using (i) and (ii)]