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Magazine Chapter 8: Problem 36
Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2}\) Find the probability that the family has the following children. No more than 4 girls
Short Answer
Expert verified
The probability is \( \frac{31}{32} \).
Step by step solution
01
Understanding the Problem
We need to calculate the probability that in a family with 5 children, there are no more than 4 girls. The binomial distribution can be used here since each child could either be a girl (G) or a boy (B), with probability 0.5 for each. This is a typical binomial probability problem where we will calculate the probability of getting 0 through 4 girls.
02
Identifying the Parameters
In this scenario, the number of trials (children) is 5, and the probability of success (having a girl) for each trial is \( p = \frac{1}{2} \). The random variable \( X \), which represents the number of girls, follows a binomial distribution: \( X \sim B(n=5, p=\frac{1}{2}) \). We need to find \( P(X \leq 4) \).
03
Calculating Individual Probabilities
We need to calculate the probabilities for \( X = 0, 1, 2, 3, \) and \( 4 \) using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \].- \( P(X = 0) = \binom{5}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^5 = \frac{1}{32} \).- \( P(X = 1) = \binom{5}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^4 = \frac{5}{32} \).- \( P(X = 2) = \binom{5}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^3 = \frac{10}{32} \).- \( P(X = 3) = \binom{5}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^2 = \frac{10}{32} \).- \( P(X = 4) = \binom{5}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^1 = \frac{5}{32} \).
04
Summing Probabilities
To find \( P(X \leq 4) \), add the probabilities calculated for \( X = 0, 1, 2, 3, \) and \( 4 \):\[P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = \frac{1}{32} + \frac{5}{32} + \frac{10}{32} + \frac{10}{32} + \frac{5}{32} = \frac{31}{32}\].
05
Conclusion
The probability that the family has no more than 4 girls is \( \frac{31}{32} \). This is almost certain, as there are only two possible gender outcomes for the 5 children, and not exceeding the count of girls by 4 is highly probable.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Theory
Probability theory is a mathematical framework for quantifying the likelihood of various outcomes. It provides us with the tools to assess and predict the probability of events, which aids in the decision-making process. In the context of this exercise, where a family has 5 children and we need to determine the probability of having no more than 4 girls, probability theory helps us calculate how likely a particular outcome is.
There are a few key principles that underpin probability theory:
There are a few key principles that underpin probability theory:
- Sample Space: This is the set of all possible outcomes. For the children in this exercise, each child is either a girl or a boy.
- Probability of an Event: This is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space. In this case, the probability that any single child is a girl is \( \frac{1}{2} \).
Random Variables
Random variables are fundamental in probability theory, as they let us express real-world phenomena in mathematical terms. A random variable can take on various outcomes, each with a specific probability. In the given problem, a random variable is used to represent the number of girls in a family of five children. This variable is discrete because it can only take on distinct values: zero to five in this case.
There are two types of random variables:
There are two types of random variables:
- Discrete Random Variables: Take on countable values, such as the number of girls among five children.
- Continuous Random Variables: Can take any value within a range, like measuring time or temperature.
Binomial Probability Formula
The binomial probability formula is essential for calculating the probability of encountering a specific number of successes in a fixed number of trials. It鈥檚 highly applicable to scenarios where there are two distinct outcomes, like having a girl or a boy.
The formula itself is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the number of trials, \( k \) is the number of successes, \( p \) is the probability of success, and \( \binom{n}{k} \) is the binomial coefficient.
In our problem, with 5 children and a 50% chance of each being a girl, the formula helps compute probabilities of having 0 through 4 girls. We then sum those probabilities to find the chance of having no more than 4 girls, arriving at the probability of \( \frac{31}{32}\). This example illustrates the power and usefulness of the binomial probability formula.
The formula itself is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the number of trials, \( k \) is the number of successes, \( p \) is the probability of success, and \( \binom{n}{k} \) is the binomial coefficient.
In our problem, with 5 children and a 50% chance of each being a girl, the formula helps compute probabilities of having 0 through 4 girls. We then sum those probabilities to find the chance of having no more than 4 girls, arriving at the probability of \( \frac{31}{32}\). This example illustrates the power and usefulness of the binomial probability formula.
