Problem 35 - Calculus for Biology and Medicine

Question

A family has four children. Assuming a \(1: 1\) sex ratio, what is the probability that no more than two children are girls?

Step-by-Step Solution

Answer

The probability that no more than two children are girls is \( \frac{11}{16} \).

Step 1: Define the Problem Context

Consider that there are four children in the family. For each child, the probability of being a girl is \( \frac{1}{2} \) and the probability of being a boy is also \( \frac{1}{2} \). This situation can be modeled using a binomial distribution, where the number of girls follows \( B(n, p) \) with \( n = 4 \) (total children) and \( p = \frac{1}{2} \) (probability of being a girl).

Step 2: Identify the Desired Probability

We need to find the probability of no more than two children being girls. This translates to the event where the number of girls \( X \) is at most 2. Mathematically, we need \( P(X \leq 2) \).

Step 3: Apply the Binomial Probability Formula

The binomial probability formula is given by:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \( \binom{n}{k} \) is the binomial coefficient, \( n = 4 \), and \( p = \frac{1}{2} \). We need to calculate \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \).

Step 4: Compute Individual Probabilities

Calculate each probability step by step:- \( P(X = 0) = \binom{4}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^4 = 1 \times 1 \times \frac{1}{16} = \frac{1}{16} \)- \( P(X = 1) = \binom{4}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^3 = 4 \times \frac{1}{2} \times \frac{1}{8} = \frac{4}{16} = \frac{1}{4} \)- \( P(X = 2) = \binom{4}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^2 = 6 \times \frac{1}{4} \times \frac{1}{4} = \frac{6}{16} = \frac{3}{8} \)

Step 5: Sum the Probabilities

To find \( P(X \leq 2) \), sum the probabilities of the individual events:\[P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = \frac{1}{16} + \frac{1}{4} + \frac{3}{8}\]Convert all fractions to a common denominator of 16:\[P(X \leq 2) = \frac{1}{16} + \frac{4}{16} + \frac{6}{16} = \frac{11}{16}\]

Key Concepts

Probability

Probability is a fundamental concept in statistics and mathematics that measures the likelihood of an event occurring. If an event is certain to happen, it has a probability of 1. If it is impossible, the probability is 0. For any event, the probability value will always lie between these two numbers.
In this exercise, we assume an equal probability of having either a boy or a girl, which is represented as a probability of \( \frac{1}{2} \). This means each time a child is born, there is a 50 percent chance the child will be a girl and a 50 percent chance the child will be a boy.
To find the probability of an event, such as having 0, 1, or 2 girls in the family of four children, we add the probabilities of each desired outcome occurring. Therefore, understanding probability helps in calculating these combined events and in predicting the likelihood of them happening.

Binomial Coefficient

The binomial coefficient is a significant part of the binomial probability formula. It is denoted by \( \binom{n}{k} \) and is used to determine the number of ways to choose \( k \) successes (like having a girl) out of \( n \) trials (like the number of children). This coefficient answers the question, "In how many different ways can we arrange a subset of a larger set?"
Mathematically, the binomial coefficient is defined as:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

In our problem, with \( n = 4 \) (the number of children) and varying \( k \) (0, 1, or 2 girls), we calculate different binomial coefficients:

• \( \binom{4}{0} = 1 \)
• \( \binom{4}{1} = 4 \)
• \( \binom{4}{2} = 6 \)

These values assist in calculating the probability of having 0, 1, or 2 girls. They simply tell us how many ways these specific numbers of girls could be arranged within four children.

Sex Ratio

The sex ratio in this exercise is assumed to be 1:1, meaning there is an equal chance of a new child being either a boy or a girl. This is a common assumption in many theoretical probability problems.
Understanding this concept helps simplify the problem, allowing it to be modeled using the binomial distribution. This equal ratio ensures that the probabilities for having each gender are balanced, and simplifies the calculations because \( p = \frac{1}{2} \).
This assumption is crucial for applying the binomial distribution as it allows the decision tree for the gender of each child to be symmetrical. The 1:1 sex ratio means that if a larger number of families each had four children, on average, they would have two girls and two boys.