Question

A bigger family. Suppose a couple will continue having children until they have at least two children of each sex (two boys and two girls). How many children might they expect to have?

Short answer

The family can expect to have 8 children in total.

Step-by-step solution

Understand the Problem

The couple will keep having children until they have at least two boys and two girls. We need to determine the expected total number of children they will have.

Set Up the Expected Value Equation

Let us denote by \(E[B]\) the expected number of boys when they have exactly two girls, and \(E[G]\) the expected number of girls when they have exactly two boys. The total expected number of children \(E\) can be expressed as the sum of the expected numbers of boys and girls: \(E = E[B] + E[G]\).

Calculate Expected Number of Boys

To find \(E[B]\), we consider that the couple will continue having children until they get two boys whenever there are exactly two girls born. The probability of the next child being a boy is \(0.5\). Define \(X\) as the number of boys a couple has to get after having two girls. Knowing they need 2 boys, \(E[X] = 2 \times (\text{expected attempts to get one boy}) \). Since the expected number of attempts before getting a single boy (alternating between possibilities of boy or girl) is \(2\), we have \(E[X] = 4\). Thus, \(E[B]=4\) boys.

Calculate Expected Number of Girls

Similarly, to find \(E[G]\), consider that the couple will continue having children until they get two girls whenever there are exactly two boys born. The expected number of attempts to have one girl is \(2\) in a series of children. Thus, for two girls, \(E[Y] = 4\). So, \(E[G] = 4\) girls.

Determine Total Expected Number of Children

Add the expected number of boys and girls to find the total expected number of children. Thus, \(E = E[B] + E[G] = 4 + 4 = 8\).

Verify the Insights

Reflect on the reasonable expectation whereby the attempts to have each gender balance out as they seek 2 of each. The final theoretical result aligns well with typical expectations in probability models for such a situation.

Key concepts

Probability

Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. It is represented as a number between 0 and 1. A probability of 0 indicates an impossible event, while a probability of 1 means the event is certain to happen.

In the context of a couple having children, we often assume the probability of having a boy is the same as having a girl, which is 0.5. This means each birth is like an independent event, unaffected by past births.

Understanding probability helps in predicting outcomes in random situations. For instance, in the problem where a couple needs two boys and two girls, probability helps us calculate expected outcomes, like the number of children they might have.

• Used to measure the chance of a possible event.
• Involves repetitive experiments or iterations.
• Essential in determining expected values.

These principles of probability allow for structured predictions based on randomness, as we see in many real-world scenarios.

Random Variable

A random variable is a variable that represents a numerical outcome of a random phenomenon. In simpler terms, it's a way to map the outcomes of a random event to numbers.

In our original exercise, the random variables are the number of boys and the number of girls the couple has until they meet their goal. These variables help in calculating quantities like expected values, which are crucial for comprehending potential outcomes and risks.

There are two types of random variables:

• Discrete Random Variables: These take on a finite or countable set of values, like the number of children in this problem.
• Continuous Random Variables: These can take any value within an interval, often seen in measurements like time or temperature.

The random variables in our problem are particularly interesting because they follow specific rules of probability, allowing for precise expectation estimates.

Binomial Distribution

Binomial distribution is a probability distribution of the number of successes in a series of independent experiments, each with the same probability of success. It's typically represented using the parameters: the number of trials (n) and the probability of success (p).

In our exercise, each birth can be seen as a trial, with a 'success' being the birth of a particular gender. The binomial distribution can model the number of boys or girls before reaching two children of each sex.

This distribution involves:

• Fixed number of trials: Here, the trials continue until certain conditions are met (two boys and two girls).
• Independent events: Each child's gender is considered an independent event with a probability of 0.5 for being a boy or girl.
• Binary outcome: Each trial results in two possible outcomes: boy or girl.

Recognizing the binomial distribution in this scenario is instrumental in calculating quantities like expected values and variances, providing insights into the likely outcomes of their family situation.