Question

Kids. A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.) a) Create a probability model for the number of children they might have. b) Find the expected number of children. c) Find the expected number of boys they'll have.

Short answer

The couple will have 2 children on average and about 0.875 boys.

Step-by-step solution

Identify Possible Outcomes for Number of Children

The couple can have a maximum of three children. Possible outcomes are: they might have 1 child (a girl on the first try), 2 children (a boy followed by a girl), or 3 children (two boys followed by a girl, or three boys). Thus, possible numbers of children are 1, 2, or 3.

Determine Probabilities for Each Outcome

For 1 child: the chance of having a girl first is \( \frac{1}{2} \).For 2 children: the chance of having a boy first then a girl is \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \).For 3 children: they might have either two boys and then a girl (\( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)) or three boys (\( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)). Thus, the probability of having 3 children is \( \frac{1}{8} + \frac{1}{8} = \frac{1}{4} \).

Construct the Probability Model

The probability model for the number of children is:- 1 child: Probability = \( \frac{1}{2} \)- 2 children: Probability = \( \frac{1}{4} \)- 3 children: Probability = \( \frac{1}{4} \)

Calculate Expected Number of Children

The expected number of children is calculated as follows: \[E(X) = 1 \times \frac{1}{2} + 2 \times \frac{1}{4} + 3 \times \frac{1}{4} = \frac{1}{2} + \frac{1}{2} + \frac{3}{4} = \frac{5}{4} + \frac{3}{4} = 2.\]Therefore, the expected number of children is 2.

Calculate Expected Number of Boys

The expected number of boys is calculated using the probability model:- For 1 child (0 boys): \( 0 \times \frac{1}{2} = 0 \).- For 2 children (1 boy): \( 1 \times \frac{1}{4} = \frac{1}{4} \).- For 3 children (2 or 3 boys): \( 2 \times \frac{1}{8} + 3 \times \frac{1}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8} \).Add these to find the expected number: \[E(B) = 0 + \frac{1}{4} + \frac{5}{8} = \frac{2}{8} + \frac{5}{8} = \frac{7}{8}.\]Thus, the expected number of boys is \( \frac{7}{8} \approx 0.875 \).

Key concepts

Expected Value

In probability and statistics, the "expected value" is a foundational concept that helps us understand the average outcome we can "expect" from a random event, over a large number of trials. It is essentially a weighted average of all possible outcomes, where each outcome is weighted by its probability of occurring. In the context of this problem, the expected value tells us the average number of children the couple might have.To find the expected number of children, we multiply the number of children by the probability of having that number of children:- For 1 child: the probability is \( \frac{1}{2} \).- For 2 children: the probability is \( \frac{1}{4} \).- For 3 children: the probability is \( \frac{1}{4} \).Thus, the expected number of children, \( E(X) \), is calculated as:\[ E(X) = 1 \times \frac{1}{2} + 2 \times \frac{1}{4} + 3 \times \frac{1}{4} = 2. \]The expected value helps us determine that, on average, the couple will have 2 children, given their stopping rules. This use of multiplication of outcomes by their probabilities to form a sum is universal in calculating expected values across different types of problems.

Probability

Probability is a way of quantifying how likely an event is to occur. It ranges from 0 to 1, where 0 means an event cannot happen, and 1 means it is certain to happen. In this exercise, we develop a probability model to evaluate the possible outcomes for the number of children the couple may have.A key aspect is understanding the individual probabilities for each outcome, such as:- The probability of having 1 child (a girl on the first try) is \( \frac{1}{2} \), because boys and girls are equally likely.- The probability of having 2 children (a boy first and then a girl) is \( \frac{1}{4} \), calculated as \( \frac{1}{2} \times \frac{1}{2} \).- The probability of having 3 children can happen in two ways: either two boys followed by a girl or three boys, each with a probability of \( \frac{1}{8} \). Adding these gives a total probability of \( \frac{1}{4} \).These outcomes are mutually exclusive, meaning only one will occur when the couple decides to have children under the specified conditions.

Statistics

Statistics provides the tools to analyze and interpret the data arising from processes having random components. In this exercise, statistics aids in shaping our understanding of the potential outcomes through the probability model and calculating expected values. Key statistical concepts applied include: - **Probability Models:** Offering a structured way to represent random events, a probability model for this scenario breaks down the likelihood of each family configuration (number of children). - **Expected Value Calculation:** By using the probabilities and corresponding outcomes, and combining them mathematically, statistics allows for calculating expected values to extrapolate averages from potential outcomes. - **Random Variables:** The number of children is a random variable, which takes on different values based on chance. In constructing the model, we categorize outcomes and determine statistical expectations based on given conditions. Understanding these statistical techniques aids in interpreting not only specific situations such as the couple's family decisions but also more complex random processes encountered in wider real-world contexts.