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?

Answer 1

In this question, first of all, draw a tabular form to write and understand the given data simply. Then calculate the probability of selecting an easy multiple-choice question and use conditional probability to get the required probability. So, use this concept to reach the solution to the given problem.
Let us define the following events as:
$E$: gets an easy question
$M$: gets a multiple-choice question
$D$: gets a difficult question
$T$: gets a True/False question
The questions in the question bank can be tabulated as follows:

| True/False| Multiple choice| Total
---|---|---|---
Easy| 300| 500| 800
Difficult| 200| 400| 600
Total| 500| 900| 1400

So, the total number of questions = 1400
Total number of multiple-choice questions = 900
Therefore, probability of selecting an easy multiple-choice question is given by
$P\left( {E \cap M} \right) = \dfrac{{500}}{{1400}} = \dfrac{5}{{14}}$
And probability of selecting a multiple-choice question is given by
$P\left( M \right) = \dfrac{{900}}{{1400}} = \dfrac{9}{{14}}$
The probability that a randomly selected question will be an easy question, given that it is a multiple-choice question is given by
$P\left( {E\left| M \right.} \right) = \dfrac{{P\left( {E \cap M} \right)}}{{P\left( M \right)}} = \dfrac{{\dfrac{5}{{14}}}}{{\dfrac{9}{{14}}}} = \dfrac{5}{9}$
Thus, the required probability is $\dfrac{5}{9}$.

Note: The probability of an event is always lying between 0 and 1 i.e., $0 \leqslant P\left( E \right) \leqslant 1$. We know that the probability of an event $E$ is given by $P\left( E \right) = \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}}$. The condition probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written as $P\left( {B\left| A \right.} \right)$.