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

The probability is \( \frac{1}{3} \).

Step 1: Understanding the Problem Situation

We have a family with two children, and we are told that at least one of them is a girl. We need to find the probability that both children are girls.

Step 2: List Possible Combinations

Since each child can either be a girl (G) or a boy (B), list all possible combinations of children for two children: GG, GB, BG, BB. However, BB is not possible since we know at least one child is a girl.

Step 3: Identify Viable Outcomes

From the possible combinations, remove the outcome 'BB' because it contradicts the condition of having at least one girl. The remaining viable outcomes are: GG, GB, and BG.

Step 4: Count Favorable Outcomes

Identify which of these outcomes meet the condition of both children being girls. Only 'GG' meets this criterion. Therefore, there is 1 favorable outcome.

Step 5: Calculate the Probability

The probability is the number of favorable outcomes divided by the number of viable outcomes. Here, it is \( \frac{1}{3} \) since there is 1 favorable outcome ('GG') and 3 viable outcomes ('GG', 'GB', 'BG').

Key Concepts

Conditional Probability

Conditional probability deals with the likelihood of an event happening given that another event has already occurred. In this exercise, knowing that at least one child is a girl changes the probability scenario. In simpler terms, we adjust our expectations based on new information.
The conditional probability formula can be stated as:

• \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

where \( P(A|B) \) is the probability of event A occurring given that B is true, \( P(A \cap B) \) is the probability of both events happening, and \( P(B) \) is the probability that event B occurs.
In this exercise, we know at least one child is a girl, changing the original equally likely set of possibilities. Instead of counting all child combinations ("GG", "GB", "BG", "BB"), we omit "BB" because it contradicts the given condition. Therefore, the probability of both being girls is calculated just from the remaining options.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting combinations and configurations. In our problem, it helps us list and count possible child gender combinations. It simplifies how we consider possibilities based on given conditions.
Each child can independently be either a girl or a boy, leading to these combinations:

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

These are considered when we don't know any detailed conditions. With the restriction of at least one girl, we eliminate "BB". This leaves us with "GG", "GB", and "BG" to consider as equally possible outcomes. By methodically listing these possibilities, combinatorics helps determine the favorable "GG" out of the three valid options.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty. This exercise focuses on calculating probability given specific conditions, showing how understanding conditions changes probabilities.
The possibility of each combination happening depends on the assumption that each child is equally likely to be a girl or boy. The theory involves determining the number of successful outcomes over the total number of viable outcomes. Thus, the formula to find our desired probability here is:

• \( \text{Probability of both being girls} = \frac{\text{Number of successful outcomes}}{\text{Number of viable outcomes}} = \frac{1}{3} \)

This shows that despite four initial combinations, the condition reduces and refines our outcome set, focusing our probability to a more specific circumstance. It also demonstrates how careful counting and condition recognition impact probability calculations.