Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The numbe

Q: Wait, why did we ignore the initial 60% female and 40% male data? Isn't that important?

That is the classic JEE trap! In probability, we must be ruthless about our sample space. The question asks for the probability *given* the candidate is from the qualified pool. The initial distribution describes the *entire* population, but once we filter for 'qualified candidates,' that initial distribution becomes irrelevant information. Always ask: 'What is my sample space?'

Q: Can I use Bayes' Theorem to solve this?

You *could*, but it would be like using a sledgehammer to crack a nut. Bayes' Theorem is powerful, but here, the problem provides the total qualified count directly. Stick to the simplest method—it reduces calculation errors and saves precious time during the exam.

Q: Why is the denominator 60 and not 100?

Because the question explicitly states: 'A candidate is randomly chosen from the qualified candidates.' This phrase restricts your universe. You are no longer picking from the 100 students; you are picking from the 60 who qualified. In probability, the denominator is always the size of the sample space you are currently operating in.

Question

Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is:

Options:

(A) $\frac{2}{3}$

(B) $\frac{11}{16}$

(C) $\frac{23}{32}$

(D) $\frac{13}{16}$

Sigma Percentile · Accepted Answer

### The Art of the Assumption: Mastering Probability

Welcome, future engineer. Today, we are going to dissect a problem that is less about complex calculus and more about the surgical precision of your mindset.

In the JEE Advanced, the most dangerous problems are not the ones with terrifying integrals, but the ones that try to lure you into a trap with unnecessary information. Let us walk through this together.

### Phase 1: The Power of the '100' Strategy

When you see percentages in a probability problem, your first instinct should be to simplify the universe. We do not know the exact number of candidates, but does it matter? No.

The probability will remain the same regardless of whether there are $100$ candidates or $1,000,000$. By assuming the total number of candidates is $100$, we turn abstract percentages into concrete integers.

This is our anchor. We have $100$ candidates, which makes our mental math effortless and keeps our focus sharp.

### Phase 2: The Trap of Irrelevance

Now, look at the data: $60\%$ female and $40\%$ male. Many students will immediately start calculating $60\%$ of $100$ and $40\%$ of $100$.

While that is mathematically correct, ask yourself: *Is it necessary?* The question asks us to pick a candidate from the **qualified** pool.

The initial gender split of the entire population is a classic distractor designed to waste your time. In the heat of the exam, recognizing what you don't need is just as important as knowing what you do.

### Phase 3: The Qualified Universe

We are told that $60\%$ of the candidates qualify. Since we assumed $100$ candidates, this means exactly $60$ candidates qualified.

This is our new universe. Our sample space is no longer $100$; it is $60$.

Every probability calculation from this point forward must be relative to this number. If you use $100$ as your denominator in the final step, you have fallen into the trap. Stay vigilant.

### Phase 4: The Algebraic Bridge

We are given a beautiful relationship: the number of qualified females is twice the number of qualified males. Let us translate this into the language of algebra.

Let the number of qualified males be $x$. Therefore, the number of qualified females is $2x$.

We know that the sum of these two groups must equal our total qualified pool of $60$. Thus, we arrive at the elegant equation:

$$3x = 60$$

Solving this is straightforward:

$$3x = 60 \Rightarrow x = 20$$

So, we have $20$ qualified males and $2 	imes 20 = 40$ qualified females. The logic holds, the numbers are clean, and we are ready for the final step.

### Phase 5: The Final Probability

Probability is simply the ratio of favorable outcomes to the total possible outcomes within the defined sample space. We want the probability that a randomly chosen qualified candidate is female.

Our favorable outcomes are the $40$ qualified females. Our total sample space is the $60$ qualified candidates. The calculation is:

$$P = \frac{40}{60} = \frac{2}{3}$$

There it is. The beauty of this problem lies in its simplicity.

It tests not just your ability to calculate, but your ability to filter noise, define your sample space, and execute with confidence. Remember this journey the next time you face a word problem. Don't just solve; understand the story the numbers are telling you. The final answer is **$2/3$**.

Select Answer:

Visualized Solution

Setting the Base

The Gender Split

Identifying the Qualified Pool

Ratio of Qualified Candidates

Forming the Equation

Solving for $x$

Calculating Qualified Females

Conditional Probability Setup

Final Probability

The Sigma Insight: Conditional Probability

The Art of the Assumption

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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