Q.Suppose that $5\%$ of men and $0.25\%$ of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Concept: Conditional Probability (Bayes' Theorem)

Let $M$ = event that the person is male, $F$ = female, $G$ = grey-haired.

Given: $P(G|M) = 0.05$, $P(G|F) = 0.0025$, and $P(M) = P(F) = 0.5$.

Step 1: Find total probability of grey hair:

$P(G) = P(M)P(G|M) + P(F)P(G|F) = 0.5 \times 0.05 + 0.5 \times 0.0025 = 0.025 + 0.00125 = 0.02625$.

Step 2: Apply Bayes' theorem:

$P(M|G) = \frac{P(M)P(G|M)}{P(G)} = \frac{0.5 \times 0.05}{0.02625} = \frac{0.025}{0.02625}$.

Step 3: Simplify:

$\frac{0.025}{0.02625} = \frac{2500}{2625} = \frac{20}{21}$.

The question asks: Given that a person has grey hair, what is the chance they are male? This is a classic conditional probability problem — we are reversing the condition. We know the probability of grey hair given gender, but we want the probability of gender given grey hair.

The natural tool here is Bayes’ theorem, which lets us “flip” conditional probabilities. But before jumping into formulas, let’s build intuition.

Imagine 1000 men and 1000 women (equal numbers).

• 5% of men have grey hair → $0.05 \times 1000 = 50$ grey-haired men.
• 0.25% of women have grey hair → $0.0025 \times 1000 = 2.5$ grey-haired women.

So total grey-haired people = $50 + 2.5 = 52.5$.

Among these, the fraction who are male is $\frac{50}{52.5} = \frac{20}{21}$. That’s the answer.

Now let’s formalise this with probability notation.

1. Define events clearly

   Let $M$ = event that the person is male, $F$ = event that the person is female, $G$ = event that the person has grey hair.

   We are given:

   • $P(G \mid M) = 5\% = 0.05$
   • $P(G \mid F) = 0.25\% = 0.0025$
   • $P(M) = P(F) = 0.5$ (equal numbers)

2. What we need

   We want $P(M \mid G)$, the probability that a grey-haired person is male.

3. Apply Bayes’ theorem

   Bayes’ theorem states:

The denominator $P(G)$ is the total probability of grey hair, found by the law of total probability:

4. Plug in the numbers

Then:

5. Simplify the fraction Multiply numerator and denominator by 10000 to clear decimals:

Divide numerator and denominator by 125:

Method: Bayes' Theorem — from "rate within a group" to "which group"

Use this when you know how common a trait is inside each group and must find, for someone who has the trait, which group they belong to.

Steps

Step 1: Record the group priors.

$P(H_i)$ is each group's share of the population (equal numbers $\Rightarrow$ each $\tfrac12$).

Step 2: Record the trait rate inside each group — mind the units.

$P(E \mid H_i)$ is the fraction of that group with the trait. Convert every percentage carefully: $0.25\%$ is $0.0025$, not $0.025$.

Step 3: Total probability of the trait.

Step 4: Bayes' theorem.

A quick route for equal group sizes: the posterior odds equal the ratio of the trait rates, so if one group's rate is $20\times$ the other's, the trait-carrier belongs to it with probability $\tfrac{20}{20+1}$.

Common Mistakes

Mistake 1: Converting $0.25\%$ to $0.025$ instead of $0.0025$.

Why it's wrong: $0.25\% = 0.25/100 = 0.0025$; the tenfold error corrupts $P(G)$ and the final ratio. Correct approach: $P(G\mid F)=0.0025$.

Mistake 2: Ignoring that the population is split equally.

Why it's wrong: the posterior weights each rate by its group's prior; only because the numbers are equal do the rates compare directly. Correct approach: $P(G)=0.5(0.05)+0.5(0.0025)=0.02625$.

Mistake 3: Finding $P(G\mid M)$ instead of $P(M\mid G)$.

Why it's wrong: the question wants the probability of being male given grey hair, the reverse direction. Correct approach: apply Bayes' theorem to get $\tfrac{20}{21}$.