Question

The family. Many couples want to have both a boy and a girl. If they decide to continue to have children until they have one child of each sex, what would the average family size be? Assume that boys and girls are equally likely.

Short answer

The average family size would be 3 children.

Step-by-step solution

Understanding the Problem

We need to calculate the average number of children a couple would have if they continue having children until they have at least one boy and one girl. This can be modeled as a probability problem where each child's gender is independent and equally likely.

Modeling the Situation

For this scenario, we'll consider a sequence of "trials" where each trial represents the birth of a child. There are precisely two outcomes: a boy (B) or a girl (G), each with probability 0.5. The process stops as soon as we have one boy and one girl.

Calculating Probabilities

The key to this problem is recognizing that the family can stop having children when they either have a series of boys followed by a girl or vice versa. An example sequence could be GBBG (2 boys and 2 girls), or BBGG (2 boys and 2 girls), and so forth.

Using Expected Value Formula

We need to find the expected value, or average, of the number of children needed. For this, compute the probabilities of all possible sequences that result in one of each gender, and weigh them by their length. Denote the expected number of trials (children) until a boy and girl are present as E.

Expectation Formula in Action

For any sequence that stops when a boy and a girl are present, consider the sequences B-G, G-B, BB-G, GG-B, BGB, GBG, etc. The probability of B-G or G-B is 0.5 * 0.5 = 0.25, with 2 children involved, thus their contribution to the expectation is 2 * 0.25 each. Sequences like BB-G and GG-B similarly involve more children but are less probable.

Solving the Expected Value Equation

Summing the infinite geometric series for expected value, considering all possibilities, it resolves to E = 1/0.5 = 2 for each gender; the total is E = 2 + 2 = 4. Thus, couples generally have around 4 children on average when aiming for at least one boy and one girl.

Key concepts

Expected Value

Expected value is a fundamental concept in probability theory, representing the average outcome if an experiment is repeated many times. In the context of our exercise, the expected value helps us determine the average family size when couples continue having children until they have both a boy and a girl.

To calculate the expected value, you consider all possible outcomes where parents stop having children: one boy and one girl. Each outcome has its own probability and number of children involved. You multiply each probability by the corresponding number of children and sum these products.

For instance, the sequences B-G or G-B both result in families with two children. Since the probability of having either is 0.25, both add up to 0.5 to the expected value. Longer sequences will add more children but occur less frequently, thus having a smaller impact.

• This process helps account for scenarios that involve more than two children in the family.
• Adding up all possible sequence probabilities gives the total expected number of children.

Understanding expected value in this problem highlights how probability theory can predict real-world scenarios.

Geometric Series

The geometric series plays a crucial role in solving our exercise by providing a method to sum infinitely many terms. In probability, this series often arises when dealing with events that repeat until a certain condition is met, such as continuing to have children until both a boy and a girl are born.

The general form of a geometric series is:\[S = a + ar + ar^2 + ar^3 + \ldots = \frac{a}{1-r}\]where \(a\) is the first term and \(r\) is the common ratio between terms. In our scenario, the geometric series allows us to calculate the expected number of trials (or births), capturing all possible ways families could achieve the desired boy-girl outcome.

Often in probability, \(r\) is a constant probability, such as 0.5 when each child being a boy or a girl is equally likely. The geometric series simplifies the process:

• Helps handle sums over infinite sequences, common in statistical predictions.
• Offers a compact way to compute probabilities and expected values across repeated trials.

This use of the geometric series consolidates the unpredictability of repeated child-gender outcomes into a manageable calculation.

Gender Prediction

Gender prediction in probability exercises refers to predicting outcomes where gender is an unknown, independent variable. In our exercise, we focused on determining the duration of childbearing until a couple has one child of each gender.

This exercise simplifies gender prediction to pure chance: equal odds for each birth being a boy or a girl. This assumption models a fair coin toss scenario, where each outcome (boy or girl) has a 50% probability.

By setting this initial probability of 0.5 for both outcomes, couples can statistically plan expectations around likely family sizes when aiming to have both gendered children using the expected value calculated:

• Provides a framework to think about gender likelihoods as independent events.
• Allows for clear probabilistic modeling of family planning scenarios.

Gender prediction, through probability theory, gives insights into managing real-life decisions around family and expectations, underlining the utility of mathematical predictions.