In a class 45% students read English, 30% read French and 20% read both English and French. One student is selected at random. Find the probability that he reads English, if it is known that he reads French.

Options

A. `1/3`
B. `2/3`
C. `5/6`
D. none of these

Answer

To solve the problem, we will use the concept of conditional probability. We want to find the probability that a student reads English given that they read French.

Let's denote:
- \( E \): the event that a student reads English
- \( F \): the event that a student reads French

We are given the following information:
- \( P(E) = 0.45 \) (45% read English)
- \( P(F) = 0.30 \) (30% read French)
- \( P(E \cap F) = 0.20 \) (20% read both English and French)

We need to find \( P(E | F) \), which is the probability that a student reads English given that they read French. According to the formula for conditional probability:

\[ P(E | F) = \frac{P(E \cap F)}{P(F)} \]

Now, we can substitute the values we have:

1. **Calculate \( P(E \cap F) \)**:
- We know \( P(E \cap F) = 0.20 \).

2. **Calculate \( P(F) \)**:
- We know \( P(F) = 0.30 \).

3. **Substitute into the conditional probability formula**:
\[ P(E | F) = \frac{P(E \cap F)}{P(F)} = \frac{0.20}{0.30} \]

4. **Simplify the fraction**:
\[ P(E | F) = \frac{20}{30} = \frac{2}{3} \]

Thus, the probability that a student reads English given that they read French is \( \frac{2}{3} \).

### Summary of the Solution Steps:
1. Identify the events and their probabilities.
2. Use the conditional probability formula.
3. Substitute the known probabilities into the formula.
4. Simplify the result to find the final probability.