Suppose that 10% of men and 5% of women have grey hair. A grey haired person is selected at random. What is the probability that the selected person is male assuming that there are 60% males and 40% females

Answer

To solve the problem step by step, we will use Bayes' theorem.

### Step 1: Define the events
Let:
- \( E_1 \): The event that the selected person is male.
- \( E_2 \): The event that the selected person is female.
- \( A \): The event that the selected person has grey hair.

### Step 2: Determine the probabilities of the events
From the problem:
- \( P(E_1) = 0.6 \) (60% of the population are males)
- \( P(E_2) = 0.4 \) (40% of the population are females)

### Step 3: Determine the conditional probabilities
We know:
- 10% of men have grey hair, so:
\[ P(A | E_1) = 0.1 \]
- 5% of women have grey hair, so:
\[ P(A | E_2) = 0.05 \]

### Step 4: Calculate the total probability of having grey hair
Using the law of total probability, we can find \( P(A) \):
\[ P(A) = P(A | E_1) \cdot P(E_1) + P(A | E_2) \cdot P(E_2) \]
Substituting the values:
\[ P(A) = (0.1 \cdot 0.6) + (0.05 \cdot 0.4) \]
Calculating this:
\[ P(A) = 0.06 + 0.02 = 0.08 \]

### Step 5: Apply Bayes' theorem to find \( P(E_1 | A) \)
We want to find the probability that the selected person is male given that they have grey hair:
\[ P(E_1 | A) = \frac{P(A | E_1) \cdot P(E_1)}{P(A)} \]
Substituting the known values:
\[ P(E_1 | A) = \frac{0.1 \cdot 0.6}{0.08} \]
Calculating this:
\[ P(E_1 | A) = \frac{0.06}{0.08} = \frac{3}{4} = 0.75 \]

### Final Answer
The probability that the selected grey-haired person is male is \( \frac{3}{4} \) or 75%.

---