Question

Out of 800 families with 4 children each, the percentage of families having no girls is:

• A. 5.25
• B. 6.25
• C. 8
• D. 12

Hint:

In binomial probability questions, first identify \(n\), \(k\), and \(p\). Remember that the total number of subjects (like 800 families) is only needed if the question asks for the 'number' of families, not the 'percentage' or 'probability'.

Answer 1

The Correct Option is B

Step 1: Concept Explanation:
This scenario aligns with a binomial distribution model, where each child's gender represents an independent event. We aim to determine the probability of a specific outcome (all boys) within a set number of events (4 children). The total number of families (800) is not needed for percentage calculation, but it is relevant if calculating the number of families fitting the criteria.

Step 2: Core Formula:
The binomial probability formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where:
- \( n \) = number of trials (children per family).
- \( k \) = number of successful outcomes (girls).
- \( p \) = probability of success (having a girl).
- \( 1-p \) = probability of failure (having a boy).

Step 3: Step-by-Step Solution:
Define the variables:
- \( n = 4 \) (4 children per family).
- \( k = 0 \) (we want families with no girls).
- \( p = 0.5 \) (probability of a girl).
- \( 1-p = 0.5 \) (probability of a boy).
Substitute into the formula: \[ P(X = 0) = \binom{4}{0} (0.5)^0 (1-0.5)^{4-0} \] \[ P(X = 0) = \frac{4!}{0!(4-0)!} \times 1 \times (0.5)^4 \] \[ P(X = 0) = 1 \times 1 \times \left(\frac{1}{2}\right)^4 \] \[ P(X = 0) = \frac{1}{16} \] Convert to percentage: \[ \text{Percentage} = \frac{1}{16} \times 100% = 6.25% \]
Step 4: Answer:
The percentage of families with no girls is 6.25%.