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: Define the possible outcomes

A family with three children can have each child as a boy (B) or a girl (G). The possible outcomes for the sexes of the three children are BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. In total, there are 8 possible combinations.

Step 2: Identify the successful outcomes

The problem asks for the probability of having at least one boy in the family. This means we identify all outcomes except the one where all children are girls (GGG). Hence, the successful outcomes are: BBB, BBG, BGB, BGG, GBB, GBG, and GGB.

Step 3: Count the total number of outcomes

The total number of outcomes is 8, as we identified in Step 1, since each child can be either a boy or a girl, resulting in these 8 different combinations.

Step 4: Calculate the number of successful outcomes

From Step 2, we identified 7 successful outcomes (BBB, BBG, BGB, BGG, GBB, GBG, GGB) where at least one child is a boy.

Step 5: Calculate the probability

The probability is the number of successful outcomes divided by the total number of possible outcomes. Therefore, the probability is \( \frac{7}{8} \).

Key Concepts

Understanding Sex Ratios

In population genetics, the sex ratio is defined as the ratio of males to females in a given population. This ratio is often expected to be 1:1, implying that there is an equal chance for a child to be a boy or a girl. This concept is fundamental in probability when determining outcomes involving human offspring.
Here, we apply the 1:1 sex ratio to predict the likelihood of various gender combinations in a group of children. Assuming each child is independently born either a boy or a girl, we are making use of the binomial theorem. This states that each child, regardless of the birth order, has a 50% chance of being a boy and a 50% chance of being a girl.
Understanding the sex ratio underpins the problem as it forms the basis of determining equal likelihoods. Incorporating this ratio assists us in confidently predicting and calculating probabilities related to sexes of children.

Basics of Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns in sets of items. In this exercise, we need combinatorics to determine all the possible outcomes when having three children.
Given that each child can be a boy (B) or a girl (G), basic combinatorial principles allow us to list all potential combinations. This is done through a method called the 'multiplication principle', which helps calculate possible scenarios by multiplying the outcomes of separate events.

• The first child has 2 outcomes: B or G.
• The second child has 2 outcomes: B or G.
• The third child also has 2 outcomes: B or G.

Multiplying these together, we get a total of 2 × 2 × 2 = 8 possible outcomes. These outcomes can be BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG.
This application of combinatorial counting determines not only the number of outcomes but lays a foundation for assessing their probabilities.

Evaluating Event Outcomes

Event outcomes in probability refer to all the possible results that can occur from an event. In this case, the event is the birth of three children within a family. We aim to determine the probability of at least one of these children being a boy.
To find this probability, we consider the successful outcomes for the event of interest. Here, successful means having at least one boy. Therefore, we systematically identify all possible outcomes that fit this criterion:

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

Each of these outcomes contains at least one boy, totaling 7 successful outcomes.
With the total number of possible outcomes being 8 (as identified using combinatorics), the probability is found by dividing successful outcomes by total outcomes. Thus, the probability of having at least one boy is \( \frac{7}{8} \).
This probability illustrates how event outcomes are assessed and connected to predictions based on defined conditions, which in this case is the 1:1 sex ratio assumption.