Problem 8 - Calculus for Biology and Medicine

Question

A family has two children. One of their children is a girl. Find the probability that both children are girls.

Step-by-Step Solution

Answer

\( \frac{1}{3} \)

Step 1: Identify Possible Outcomes

The possible outcomes for the two children's genders are: Boy-Boy (BB), Boy-Girl (BG), Girl-Boy (GB), and Girl-Girl (GG). We know at least one child is a girl.

Step 2: Remove Improbable Outcome

Since one child is a girl, the outcome BB is impossible. The remaining outcomes are BG, GB, and GG.

Step 3: Calculate Probability of Both Being Girls

Only the outcome GG meets the condition that both are girls. Out of the three possible outcomes (BG, GB, GG), GG is one, so the probability is \( \frac{1}{3} \).

Key Concepts

Conditional Probability

In probability, we often deal with situations where we have some prior information that affects the likelihood of different outcomes. This is known as conditional probability. It's the probability of an event occurring, given that another event has already happened. In our problem where a family has two children, we know that one of them is a girl. This information is crucial because it affects how we analyze the possible gender combinations of the children.
To calculate the conditional probability, we focus only on those outcomes that satisfy the given condition. Here, once it’s known that at least one child is a girl, the possible combinations are narrowed down:

• Boy-Girl (BG)
• Girl-Boy (GB)
• Girl-Girl (GG)

So, the probability of both children being girls, given that one is a girl, is determined from this subset of outcomes, not the original set. Thus, understanding conditional probability helps in appropriately focusing on relevant outcomes.

Sample Space

The concept of a sample space is fundamental in probability. It consists of all the possible outcomes of an experiment. In the context of the given problem, the sample space represented all combinations of genders for two children. Originally, these possible outcomes were:

• Boy-Boy (BB)
• Boy-Girl (BG)
• Girl-Boy (GB)
• Girl-Girl (GG)

The sample space provides a complete listing of what could occur before any information is received. However, once you know a particular condition (like one child being a girl), the relevant sample space effectively shrinks. This narrowing is crucial for accurately determining relevant probabilities. The concept of a sample space allows you to see all initial possibilities and aids in clear outcome analysis.

Outcome Analysis

Outcome analysis is pivotal when dissecting any probability problem. This involves examining each potential occurrence within the sample space, often influenced by given conditions. For the problem at hand, outcome analysis starts by identifying original possible gender arrangements for two children.
Once informed that at least one child is a girl, it eliminates the BB outcome. The refined set of outcomes (BG, GB, GG) must be scrutinized.
Through this analysis, we evaluate how many of these outcomes meet the condition where both children are girls. Only GG fits this description, making 1 of the 3 available options valid. This methodical examination confirms the probability of both children being girls as \( \frac{1}{3} \).
Thus, outcome analysis enables a detailed look at each possibility, ensuring accuracy and understanding of probability questions.