3. In a certain class 25% of the students failed in Mathematics, 15% failed in chemistry, 10% failed in
both mathematics & chemistry. A student is selected at random.
i) If he failed in Mathematics, what is the probability that he failed in chemistry?
ii) If he failed in chemistry, what is the probability that he failed in mathematics?

This problem can be solved using the concept of conditional probability and the principle of inclusion-exclusion. Let’s define the following events:

- \( M \): The event that the student failed in Mathematics.
- \( C \): The event that the student failed in Chemistry.

We are given the following probabilities:
- \( P(M) = 0.25 \) (25% of students failed in Mathematics),
- \( P(C) = 0.15 \) (15% of students failed in Chemistry),
- \( P(M \cap C) = 0.10 \) (10% of students failed in both Mathematics and Chemistry).

Now, let's solve each part.

### Part (i): If the student failed in Mathematics, what is the probability that he also failed in Chemistry?

We need to find \( P(C|M) \), the conditional probability that a student failed in Chemistry given that they failed in Mathematics.

Using the formula for conditional probability:
\[
P(C|M) = \frac{P(M \cap C)}{P(M)}
\]

Substitute the known values:
\[
P(C|M) = \frac{0.10}{0.25} = 0.40
\]

So, the probability that a student who failed in Mathematics also failed in Chemistry is 0.40 or 40%.

### Part (ii): If the student failed in Chemistry, what is the probability that he also failed in Mathematics?

We need to find \( P(M|C) \), the conditional probability that a student failed in Mathematics given that they failed in Chemistry.

Again, using the formula for conditional probability:
\[
P(M|C) = \frac{P(M \cap C)}{P(C)}
\]

Substitute the known values:
\[
P(M|C) = \frac{0.10}{0.15} = 0.67
\]

So, the probability that a student who failed in Chemistry also failed in Mathematics is approximately 0.67 or 67%.

### Summary:
1. \( P(C|M) = 0.40 \)
2. \( P(M|C) = 0.67 \)

Would you like more detailed explanations, or do you have any questions?

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Here are 5 related questions to consider:
1. How do we interpret conditional probability in real-world scenarios?
2. What is the principle of inclusion-exclusion, and how was it applied here?
3. Can conditional probability exceed 1? Why or why not?
4. How would these probabilities change if the intersection of failures in both subjects was different?
5. What if a student was selected randomly from a group that passed Mathematics? What would the conditional probabilities be?

**Tip**: Conditional probability is particularly useful when additional information is known, as it adjusts the probability based on that new knowledge!

In a certain class 25% of the students failed in Mathematics, 15% failed in Chemistry, 10% failed in both Mathematics and Chemistry. A student is selected at random. i) If he failed in Mathematics, what is the probability that he failed in Chemistry? ii) If he failed in Chemistry, what is the probability that he failed in Mathematics?

Explore the concept of conditional probability through a real-world problem where 25% of students failed in Mathematics, 15% in Chemistry, and 10% in both. Learn how to calculate P(C|M) and P(M|C) using key probability formulas. This is suitable for students in Grades 10-12.

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