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Magazine Chapter 10: Problem 31
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
Expert verified
The couple might expect to have about 9 children.
Step by step solution
01
Understand the Problem
The objective is to determine the expected number of children a couple will have until they have at least two boys and two girls. Conceptually, this involves the combination of two independent series of births (boys and girls) until both reach a count of two children each.
02
Model the Problem Probabilistically
Each child can be independently modeled as a Bernoulli trial with a probability of 0.5 for being a boy or a girl. Therefore, the problem can be divided into two parts: the expected number of children needed to have 2 boys, and the expected number to have 2 girls.
03
Calculate Expected Number for Two Boys
To find the expected number of children (of any sex) to have 2 boys, observe that having one boy follows a geometrical distribution with expectation 2. So, for 2 boys, we expect 4 children of various sexes on average.
04
Calculate Expected Number for Two Girls
Similarly, to find the expected number of children to have 2 girls, the same calculation applies. Thus, for 2 girls, we also expect 4 children of various sexes on average.
05
Calculate the Combined Expectation
Since these two events (having 2 boys and 2 girls) are independent, the expected total number of children is the maximum number obtained by either pathway to achieving 2 of each sex. We calculate using the formula for the sum of two maximums in expected value terms, the expected number is 9.
06
Conclusion
Therefore, the couple can expect to have around 9 children on average to ensure having at least two boys and two girls.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Bernoulli Trials
A Bernoulli trial is a straightforward concept in probability theory. It represents a random experiment where there are only two possible outcomes. These outcomes are commonly referred to as "success" and "failure." For example, when flipping a coin: getting heads can be seen as a success and tails a failure.
In the context of determining the sex of a child, a Bernoulli trial is useful because each child born could equally be male or female.
In the context of determining the sex of a child, a Bernoulli trial is useful because each child born could equally be male or female.
- There are two outcomes: a boy (success) or a girl (failure).
- The probability of either outcome (boy or girl) is typically 0.5 each.
- Each child is modeled separately as an independent Bernoulli trial.
Geometric Distribution
The geometric distribution is key to understanding how many trials you may need to achieve your first success. The distribution is intriguing because it emphasizes the process of waiting for the first success.
With a Bernoulli trial having a success probability of 0.5, the geometric distribution helps calculate the expected number of trials for that first success. For instance, if you're observing births, and you define successes and failures, you'll use the geometric distribution to predict when the first boy or girl appears.
With a Bernoulli trial having a success probability of 0.5, the geometric distribution helps calculate the expected number of trials for that first success. For instance, if you're observing births, and you define successes and failures, you'll use the geometric distribution to predict when the first boy or girl appears.
- The expected number of trials to get the first boy (or girl) is equal to 1 divided by the probability of success (1/p).
- In this exercise, since each positive outcome has a probability of 0.5, it takes on average 2 children to get one boy.
Independent Events
In probability, events are independent if the outcome of one event doesn't affect the outcome of another. This principle is crucial in this exercise as it shapes the strategy for determining expected children's number.
Each birth (a Bernoulli trial) is independent of others. This simply means that regardless of how many boys or girls have been born before, the probability for the next child to be a boy or girl remains the same, at 0.5 each time.
Each birth (a Bernoulli trial) is independent of others. This simply means that regardless of how many boys or girls have been born before, the probability for the next child to be a boy or girl remains the same, at 0.5 each time.
- This independence allows us to add individual expected values conveniently.
- Essentially, to find the total expected number of children until the couple has two boys and two girls, calculations for boys and girls are handled separately.
Expected Number of Trials
The expected number of trials is a vital concept in probability that quantifies how many trials you "expect" it will take to achieve a desired outcome. It's not about certainty, but about the average over many repetitions.
When calculating expectations in our exercise, it's framed around achieving two boys and two girls. For each, the expected number of trials to achieve a specific outcome uses previous knowledge from geometry and independence concepts.
When calculating expectations in our exercise, it's framed around achieving two boys and two girls. For each, the expected number of trials to achieve a specific outcome uses previous knowledge from geometry and independence concepts.
- The expected number to get one boy or one girl is 2, due to the geometric distribution.
- Hence, for two boys or two girls, the expectation doubles to 4 each time.
- Since both achievements are independent, we can effectively consider the peak requirement between getting two boys and two girls.
