Problem 11 - An Introduction to Mathematical Statistics and Its Applications

Question

If a family has four children, is it more likely they will have two boys and two girls or three of one sex and one of the other? Assume that the probability of a child being a boy is \(\frac{1}{2}\) and that the births are independent events.

Step-by-Step Solution

Answer

Depending on solution values obtained, specify if it's more likely that the family will have two boys and two girls or three children of one sex and one of the other based on which probability is larger.

Step 1: Probability of two boys and two girls

First, we need to calculate the probability of the family having exactly two boys and two girls. This can be done using the formula for binomial probability: \[P(X=k)=\binom{n}{k}*(p^k)*(1-p)^{n-k}\] where \(n\) is the number of trials or events, \(k\) is the number of 'success' events, \(p\) is the probability of success on each trial and \(\binom{n}{k}\) is the binomial coefficient, also interpreted as the different ways of choosing \(k\) successes from \(n\) trials. Here, 'success' is defined as having a boy, and 'failure' as having a girl. Plugging the values \(n=4\), \(k=2\), and \(p=\frac{1}{2}\), we get: \[P(2 boys and 2 girls)=\binom{4}{2}*(0.5)^2*(1-0.5)^{4-2}\]

Step 2: Probability of three of one sex and one of the other

The probability of having three children of one sex and one of the other would imply either three boys and one girl or one boy and three girls. These are two distinct events, so we must calculate the probability of each and add them together. Similar to above, we plug in \(n=4\), and \(p=\frac{1}{2}\), but this time for \(k=3\) and \(k=1\), resulting in: \[P(3 boys and 1 girl)=\binom{4}{3}*(0.5)^3*(1-0.5)^{4-3}\] and \[P(1 boy and 3 girls)=\binom{4}{1}*(0.5)^1*(1-0.5)^{4-1}\] The total probability of having three of one sex and one of the other is the sum of these two probabilities.

Step 3: Comparing the probabilities

Once calculated, compare the two probabilities from Step 1 and Step 2. If the probability from Step 1 is larger, it's more likely that the family will have two boys and two girls. If the probability from Step 2 is larger, it's more likely that the family will have three children of one sex and one of the other.

Key Concepts

Probability Theory

Probability theory helps us understand the likelihood of different outcomes in uncertain situations, like the birth of children in a family.
In probability, we define certain terms such as 'trial', which refers to an action or experiment, like the birth of a child.
The result of each trial is called an 'outcome', and the set of all possible outcomes is known as the 'sample space'.In this exercise, we use the concept of binomial probability. The formula for binomial probability is \[P(X=k)=\binom{n}{k}(p^k)(1-p)^{n-k}\] where:
- \(n\) is the total number of trials (here, the total number of children).
- \(k\) is the number of times you want your outcome (like having two boys).
- \(p\) is the probability of achieving your desired outcome on a single trial (here \( \frac{1}{2} \) for a boy).
- \(1-p\) is the probability of the other outcome (for a girl, also \( \frac{1}{2} \)).
Using this theory, we calculate how likely it is to have different numbers of boys and girls in a family with four children.

Independence of Events

The independence of events is a crucial idea in probability theory. When events are independent, the outcome of one event does not affect the outcome of another.
In simpler terms, what happened before doesn’t affect what happens next. In this family example, the birth of each child is an independent event. That means each birth has a separate and unchanged fifty percent chance of being a boy or a girl, regardless of the gender of the previous children.
This concept makes it easier to calculate probabilities, as you don't have to adjust for previously occurring events. Since each birth is independent, it allows us to use the binomial probability formula reliably. Other than family scenarios, this principle can be applied to many other events, like flipping a coin or rolling a dice, where past results have no bearing on future attempts.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects.
It's used extensively in probability to find out how many different ways an outcome can occur.In this problem, we use combinatory calculations to determine how many ways we can have certain numbers of boys and girls in the family. The binomial coefficient, shown as \(\binom{n}{k}\), is a combinatorial calculation that tells us the number of ways to choose \(k\) successes (boys) from \(n\) trials (children).For example:
- \(\binom{4}{2}\) calculates the number of ways to have two boys in four children.
- \(\binom{4}{3}\) calculates the different ways to have three boys in four children.
These coefficients are used within the binomial probability formula to account for all the ways an outcome can occur, which then helps determine the overall probability of that outcome.