Problem 34 - Calculus for Biology and Medicine

Question

A family has three children. Assuming a \(1: 1\) sex ratio, what is the probability that at least one child is a boy?

Step-by-Step Solution

Answer

The probability that at least one child is a boy is \( \frac{7}{8} \).

Step 1: Understanding the Problem

We have a family with three children, and we need to determine the probability that at least one of them is a boy.

Step 2: Identifying Possible Outcomes

Each child can be either a boy (B) or a girl (G), leading to the sample space having possible outcomes like BBB, GBB, BGB, etc. There are a total of \[2^3 = 8\] possible combinations of children.

Step 3: Counting Favorable Outcomes

All outcomes except GGG include at least one boy. Thus, the favorable outcomes are BBB, BBG, BGB, BGG, GBB, GBG, and GGB. That totals to 7 outcomes.

Step 4: Calculating the Probability

The probability of an event is calculated using \[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]Substitute the values: \[ P = \frac{7}{8} \]

Step 5: Verification Using Complementary Probability

Instead of counting the favorable outcomes, calculate the probability of the complementary event (all girls, GGG) and subtract from 1:\[ P(\text{at least one boy}) = 1 - P(\text{all girls}) = 1 - \frac{1}{8} = \frac{7}{8} \]

Key Concepts

Understanding Sample Space

In probability theory, a sample space is a critical concept. It represents the set of all possible outcomes of a particular experiment or scenario. For the family with three children example, each child can either be a boy (B) or a girl (G), which leads to a binary choice for each child.

When we calculate the sample space for three children, we are looking at every possible combination of boys and girls across those children. Since each child can be one of two genders, the total number of possible combinations can be calculated with the formula \[2^3 = 8\].

The sample space in this scenario includes:

• BBB — All children are boys
• BBG — First two are boys, last is a girl
• BGB — Boy, then girl, then boy
• BGG — Boy, followed by two girls
• GBB — Girl, followed by two boys
• GBG — Girl, boy, girl
• GGB — First two are girls, last is a boy
• GGG — All children are girls

Understanding the sample space is fundamental to solving probability problems as it defines all the potential outcomes.

Counting Favorable Outcomes

Favorable outcomes are the set of outcomes that satisfy the condition we are interested in. In this problem, we are interested in finding the probability of having at least one boy among three children.

To determine the favorable outcomes, we need to identify which combinations include at least one boy. From the sample space of eight possible outcomes, all combinations except "GGG" contain at least one boy. Thus, the favorable outcomes include:

• BBB
• BBG
• BGB
• BGG
• GBB
• GBG
• GGB

There are seven such combinations, making these the favorable outcomes. By counting these, we can move forward in calculating the probability of this event.

Exploring Complementary Probability

Complementary probability refers to the concept where the probability of an event occurring is related to the probability of the event not occurring. Specifically, the sum of the probability of an event and its complement (the event not happening) is always 1.

In our scenario, we want to find the probability of having at least one boy. An easier way sometimes is to first consider the complementary event, which is having no boys, or all girls. In our sample space, "GGG" is the only outcome where all the children are girls.

The probability of this complementary event (all girls) is calculated as follows: \[P(\text{all girls}) = \frac{1}{8}\]

Using complementary probability, we subtract this result from 1:\[P(\text{at least one boy}) = 1 - P(\text{all girls}) = 1 - \frac{1}{8} = \frac{7}{8}\]

This further verifies our earlier calculation and shows that complementary probability offers another reliable way to solve probability problems.