Q.In a school, there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in Class XII given that the chosen student is a girl?

The key idea is conditional probability: the probability of event $A$ (studies in Class XII) given event $B$ (student is a girl) is $P(A|B) = \frac{P(A \cap B)}{P(B)}$.

Step 1: Total students = 1000, total girls = 430. So $P(B) = \frac{430}{1000}$. …

The core idea here is conditional probability — the chance of one event happening when we already know another event has occurred. When we say "given that the chosen student is a girl," we are restricting our universe to only the girls. The probability we want is: out of all girls, what fraction are in Class XII?

This is not about the whole school of 1000 students. The "given" condition shrinks the sample space. So we don't need the total number of students at all — only the number of girls and the number of girls in Class XII.

Let’s work through it step by step.

1. Define the events clearly.

   Let $A$ be the event that a randomly chosen student studies in Class XII.

   Let $B$ be the event that the chosen student is a girl.

   We are asked for $P(A \mid B)$, the probability of $A$ given $B$.

2. Recall the formula for conditional probability.

Here $A \cap B$ is the event that the student is both a girl and studies in Class XII.

3. Extract the given numbers.

   Total students = 1000.

   Number of girls = 430.

   Out of these 430 girls, 10% study in Class XII.

   So number of girls in Class XII = $10\%$ of $430 = \frac{10}{100} \times 430 = 43$.

4. Compute the probabilities.

5. Apply the conditional probability formula. $P(A \mid B) = \frac{0.043}{0.43} = \frac{43}{430} = 0.1$ …

Method: Conditional probability as a proportion within the condition group

Use this whenever "given that the item belongs to group $B$" is really asking for a fraction inside that group. It is the fastest reading of conditional probability for count/percentage data.

Steps

Step 1: Treat the conditioning group as the new whole.

"Given the student is a girl" means the girls become $100\%$ of the world you now measure against. The overall population total stops mattering.

Step 2: Take the favourable count over the group count.

Here that is (girls in Class XII) $\div$ (all girls). …

Common Mistakes

Mistake 1: Keeping the whole school of $1000$ in the denominator after conditioning.

Why it's wrong: "given the student is a girl" restricts the world to the $430$ girls, so the denominator must be $430$. Correct approach: compute (girls in Class XII)$/$(all girls) $=\tfrac{43}{430}=\tfrac{1}{10}$.

Mistake 2: Misreading "$10\%$ of the $430$ girls" as $10\%$ of all students. …