A couple has four children. Assuming that the probability of a boy and a girl are equal and each child's gender is independent, what is the probability that the family has at least one girl?

Introduction / Context:
This question concerns a family with four children and asks for the probability of having at least one girl. It is a classic example of using complementary probability in problems involving repeated independent trials, with each trial having two equally likely outcomes (boy or girl).

Given Data / Assumptions:

• The couple has four children.
• Each child is equally likely to be a boy or a girl.
• Gender outcomes for different children are independent.
• We want the probability of having at least one girl among the four children.

Concept / Approach:
Calculating the probability of "at least one girl" directly would require summing probabilities for one girl, two girls, three girls, and four girls, which is possible but lengthy. A more efficient approach is to use the complement: first compute the probability of having no girls (that is, all boys) and then subtract that from 1. This method is standard for "at least one" type questions.

Step-by-Step Solution:
Probability that one child is a boy = 1/2, and girl = 1/2. Event of interest: at least one girl among four children. Complement event: no girls among four children, that is, all four are boys. Probability that one child is a boy = 1/2. Probability that all four are boys = (1/2) ^ 4 = 1 / 16. Required probability = 1 minus probability of all boys. Compute 1 - 1/16 = 15/16. Convert 15/16 to decimal: 15 divided by 16 = 0.9375.

Verification / Alternative check:
We can expand the binomial distribution for four independent trials. The total probability across all possibilities (0, 1, 2, 3, 4 girls) must sum to 1. The case of 0 girls has probability (1/2)^4 = 1/16. Therefore, the sum of probabilities for 1 or more girls is 1 - 1/16, which again is 15/16. This matches the earlier calculation, confirming our result of 0.9375.

Why Other Options Are Wrong:
0.0625: This is 1/16 and represents the probability of all boys, not at least one girl.
0.5: This would mean that half of all such families have at least one girl, which is too small because only a small fraction have no girls.
0.0257: This value does not correspond to any standard fraction in this context and is far from the correct value.
None of these: This is incorrect because 0.9375 is present in the options and matches the correct probability.

Common Pitfalls:
A common error is to interpret "at least one girl" as "exactly one girl". Another pitfall is to try to add many separate cases directly instead of using the complement rule. Some learners also incorrectly assume that four children automatically mean two boys and two girls, which ignores the probabilistic nature of the problem.

Final Answer:
Hence, the probability that the couple with four children has at least one girl is 0.9375.