Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Gender Makeup of a Family 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. (picture cannot copy) No more than 4 girls

Step-by-Step Solution

Answer

The probability that the family has no more than 4 girls is \(\frac{31}{32}\).

Step 1: Understanding the Binomial Distribution

In this problem, the number of children in the family (each child being either male or female) can be treated as a binomial experiment. Each individual child can be either a girl (success) or a boy (failure) with probability \(p = \frac{1}{2}\). The number of trials, \(n\), is 5 since there are 5 children.

Step 2: Identifying "No More Than 4 Girls"

The phrase "no more than 4 girls" means that the possible outcomes include having 0, 1, 2, 3, or 4 girls. We need to calculate the probability of these cases and sum them up.

Step 3: Binomial Probability Formula

The probability of getting exactly \(k\) successes (girls, in this case) in \(n\) trials is given by the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

Step 4: Probability for Each Outcome

Calculate the probability of getting 0, 1, 2, 3, and 4 girls using the formula:- \(P(0\text{ girls}) = \binom{5}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^5 = \frac{1}{32}\)- \(P(1\text{ girl}) = \binom{5}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^4 = \frac{5}{32}\)- \(P(2\text{ girls}) = \binom{5}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^3 = \frac{10}{32}\)- \(P(3\text{ girls}) = \binom{5}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^2 = \frac{10}{32}\)- \(P(4\text{ girls}) = \binom{5}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^1 = \frac{5}{32}\)

Step 5: Summing the Probabilities

Add the probabilities calculated in Step 4 together to find the probability of having no more than 4 girls:\[ P(X \leq 4) = P(0) + P(1) + P(2) + P(3) + P(4) = \frac{1}{32} + \frac{5}{32} + \frac{10}{32} + \frac{10}{32} + \frac{5}{32} = \frac{31}{32} \]

Key Concepts

Probability Theory

Probability theory is the mathematical framework that helps us understand and predict random events. It's like a set of rules for how likely different outcomes are, under given conditions. In the family example, each child born represents a random event with a probability of being a girl or a boy. The probability, indicated as \( p \), is \( \frac{1}{2} \) for each outcome, meaning there is an equal chance of having a boy or a girl. This simplicity helps in calculating multiple child outcomes.

• Event: An occurrence like having a girl.
• Probability of an Event: The chance that a particular event occurs, between 0 and 1.
• Random Experiment: Repeating an event, like having children, several times to see different outcomes.

Probability theory allows us to combine these understood probabilities to predict how likely specific outcomes are, which is expressed using formulas, like the binomial probability formula.

Binomial Coefficient

The binomial coefficient plays a crucial role in calculating probabilities in situations like the family example. The binomial coefficient, denoted as \( \binom{n}{k} \), helps determine how many ways \( k \) successes (girls, here) can occur in \( n \) total events (children).
To calculate the binomial coefficient, use the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( n! \) represents \( n \) factorial, the product of all positive integers up to \( n \). For example, \( \binom{5}{3} = \frac{5!}{3! \, 2!} = 10 \), indicating there are 10 ways to have 3 girls out of 5 children.

• Selection of Events: Number of ways to pick successes from trials.
• Facilitates Probability Computation: Used in the binomial distribution formula.

Understanding and applying the binomial coefficient is essential in breaking down complex probability problems into manageable calculations.

Binomial Experiment

A binomial experiment is a statistical experiment that satisfies certain criteria, making it suitable for binomial distribution calculations. In our family example, each child's birth is a trial with two possible outcomes. The binomial experiment checks these conditions:

• Fixed Number of Trials (\( n \)): Here, 5 children or trials.
• Two Possible Outcomes: Success (girl) or failure (boy).
• Constant Probability: Each child has a probability \( p = \frac{1}{2} \) for being a girl.
• Independent Trials: Birth outcomes of individual children don't influence each other.

These criteria establish that the family scenario is a binomial experiment. This scenario can then be analyzed using the binomial probability formula, allowing us to compute the probability of different numbers of girls in the family of 5 children. Binomial experiments are powerful as they simplify the analysis of repeated events that have binary outcomes.