Two parents plan to have three children. What is the probability that the children will be two girls and one boy?

Verified step by step guidance

1. Identify the total number of children and the desired gender combination: 3 children with exactly 2 girls and 1 boy.
2. Recognize that each child’s gender is an independent event with a probability of 1/2 for a girl and 1/2 for a boy.
3. Calculate the number of different ways to arrange 2 girls and 1 boy among 3 children using combinations: $C(3,2)=3$.
4. Determine the probability of one specific arrangement (e.g., girl, girl, boy) by multiplying the probabilities: ${\frac{1}{2}}^2\backslash times\frac{1}{2}$.
5. Multiply the number of arrangements by the probability of one arrangement to get the total probability: $C(3,2)\backslash times{\frac{1}{2}}^2\backslash times\frac{1}{2}$.

Key Concepts

Basic Probability

Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1. In genetics, it helps predict the chances of offspring having certain traits, such as gender, assuming each outcome is equally likely and independent.

Binomial Probability Distribution

This distribution calculates the probability of a specific number of successes (e.g., girls) in a fixed number of independent trials (e.g., children), each with the same probability of success. It uses combinations to account for different orders of outcomes.

Combinatorics (Permutations and Combinations)

Combinatorics involves counting the number of ways events can occur. For this problem, combinations determine how many ways two girls and one boy can be arranged among three children, which is essential for calculating the total probability.