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, then the probability that it will be an easy question given that it is a multiple choice question is:

• A. $\frac{8}{14}$
• B. $\frac{9}{14}$
• C. $\frac{5}{14}$
• D. $\frac{5}{9}$

Answer 1

The correct option is D $\frac{5}{9}$

Here there are two types of questions, True/False or multiple choice questions (T/F or MCQ) and each of them is divided into easy and difficult type, as shown below in the tree diagram.



So, in all, there are $500$ T/F questions and $900$ MCQ's
Also, there are $800$ easy questions and $600$ difficult questions.
The sample space of the experiement 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'
$\Rightarrow P\left(E\right)=\frac{800}{1400}=\frac{8}{14}$ and $P\left(F\right)=\frac{900}{1400}=\frac{9}{14}$
Now, $E\cap F$ is the event of getting an MCQ which is easy.
Clearly, from the diagram, we know that there are $500$ MCQ's that are easy.
So $P(E\cap F)=\frac{500}{1400}=\frac{5}{14}$
Now, we know that by definition of conditional probability
$P\left(E/F\right)=\frac{P(E\cap F)}{P\left(F\right)}$
$\Rightarrow P\left(E/F\right)=\frac{5/14}{9/14}=\frac{5}{9}$