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: Understanding the Concept:
This problem can be modeled using a binomial distribution. Each child's gender is an independent trial. We are looking for the probability of a specific outcome (no girls) in a fixed number of trials (4 children). The number of families (800) is extra information not needed to calculate the percentage, but would be needed to find the number of families.

Step 2: Key Formula or Approach:
The binomial probability formula is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where:
- \( n \) is the number of trials (number of children in a family).
- \( k \) is the number of successful outcomes (number of girls).
- \( p \) is the probability of success on a single trial (probability of having a girl).
- \( 1-p \) is the probability of failure (probability of having a boy).

Step 3: Detailed Explanation:
Let's define the parameters for this problem:
- The number of trials, \( n \), is the number of children in each family, so \( n = 4 \).
- We are interested in families having no girls, so the number of successes, \( k \), is 0.
- The probability of having a girl, \( p \), is assumed to be 0.5.
- The probability of having a boy, \( 1-p \), is also 0.5.
Now, we substitute these values into the binomial formula: \[ P(X = 0) = \binom{4}{0} (0.5)^0 (1-0.5)^{4-0} \] \[ P(X = 0) = \frac{4!}{0!(4-0)!} \times 1 \times (0.5)^4 \] \[ P(X = 0) = 1 \times 1 \times \left(\frac{1}{2}\right)^4 \] \[ P(X = 0) = \frac{1}{16} \] To express this probability as a percentage, we multiply by 100: \[ \text{Percentage} = \frac{1}{16} \times 100% = 6.25% \]
Step 4: Final Answer:
The percentage of families having no girls is 6.25%.