Assume that the birth of a boy or girl to a couple to be equally likely, mutually exclusive, exhaustive and independent of the other children in the family. For a couple having `6` children, the probability that their "three oldest are boys" is

Options

A. `(20)/(64)`
B. `(1)/(64)`
C. `(8)/(64)`
D. none of these

Explanation

To solve the problem, we need to find the probability that the three oldest children in a family of six are boys.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a couple with 6 children, and we want to find the probability that the three oldest children are boys. The birth of a boy or girl is equally likely.

2. **Defining the Events**:
Let:
- B = Boy
- G = Girl

The probability of having a boy (P(B)) = 1/2 and the probability of having a girl (P(G)) = 1/2.

3. **Fixing the First Three Children**:
Since we want the three oldest children to be boys, we can fix the first three children as boys:
- Child 1: B
- Child 2: B
- Child 3: B

The probability of this event (the first three being boys) is 1, as it is a fixed condition.

4. **Calculating the Probability for the Remaining Children**:
For the remaining three children (Child 4, Child 5, and Child 6), each child can either be a boy or a girl. The probability for each of these children is still 1/2.

Therefore, the probability for each of the last three children can be calculated as:
- P(Child 4) = 1/2
- P(Child 5) = 1/2
- P(Child 6) = 1/2

5. **Calculating the Total Probability**:
Since the events are independent, we multiply the probabilities of the last three children:
\[ P(\text{Child 4, Child 5, Child 6}) = P(Child 4) \times P(Child 5) \times P(Child 6) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

6. **Final Probability**:
The total probability that the three oldest children are boys and the remaining three can be either boys or girls is:
\[ P(\text{Three oldest are boys}) = 1 \times \frac{1}{8} = \frac{1}{8} \]

### Conclusion:
Thus, the probability that the three oldest children are boys is \( \frac{1}{8} \).