Q.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?

The key idea is conditional probability: we restrict the sample space to only multiple choice questions and find the fraction of those that are easy.

Step 1 – Total questions in the bank

Easy T/F: 300, Difficult T/F: 200, Easy MCQ: 500, Difficult MCQ: 400.

Total = $300 + 200 + 500 + 400 = 1400$.

Step 2 – Restrict to multiple choice questions

Total MCQ = Easy MCQ + Difficult MCQ = $500 + 400 = 900$. …

The core idea here is conditional probability: the probability of event $A$ happening given that event $B$ has already occurred. The formula is:

In this problem, we want $P(\text{Easy} \mid \text{Multiple Choice})$. That means we restrict our attention only to multiple choice questions, and then ask: what fraction of those are easy?

Let’s break it down step by step.

1. Organize the data.

   The question bank has four categories. Let’s list them clearly:

   Type	Easy	Difficult	Total
   True/False	300	200	500
   Multiple Choice	500	400	900
   Total	800	600	1400

   The total number of questions is $300 + 200 + 500 + 400 = 1400$.

2. Identify the events.

   Let $E$ be the event that a question is easy.

   Let $M$ be the event that a question is multiple choice.

   We need $P(E \mid M)$.

3. Find $P(M)$ — the probability that a randomly selected question is multiple choice.

   From the table, the total number of multiple choice questions is $500 + 400 = 900$.

   So:

4. Find $P(E \cap M)$ — the probability that a question is both easy and multiple choice. The number of easy multiple choice questions is $500$. So:

5. Apply the conditional probability formula. …

Method: Conditional probability from a two-way frequency table

Use this whenever items are cross-classified by two attributes (here difficulty $\times$ question-type) and you must find the probability of one attribute given a value of the other.

Steps

Step 1: Organise the counts into a two-way table.

Put one attribute across the columns and the other down the rows, and fill in the row totals, column totals, and grand total. This lays out every count the formula needs.

Step 2: Identify the conditioning category = the relevant margin.

"Given that it is a multiple-choice question" means you work inside the multiple-choice column only. Its total is the denominator:

Step 3: Take the single cell over that margin total. …

Common Mistakes

Mistake 1: Dividing by the full total of $1400$.

Why it's wrong: "given that it is a multiple choice question" restricts the world to the $900$ MCQs, so the denominator must be $900$, not $1400$. Correct approach: $P(\text{Easy}|\text{MCQ})=\frac{500}{900}=\frac59$.

Mistake 2: Computing $P(\text{MCQ}|\text{Easy})$ by mistake. …