In Exercises 29–32, use the given sample space or construct the required sample space to find the indicated probability.




Four Children Exercise 29 lists the sample space for a couple having three children. After identifying the sample space for a couple having four children, find the probability of getting three girls and one boy (in any order).

Verified step by step guidance

1. Step 1: Understand the problem. The couple has four children, and we need to identify the sample space for all possible gender combinations of these children. Each child can either be a boy (B) or a girl (G), so the sample space consists of all possible arrangements of B and G for four children.
2. Step 2: Construct the sample space. Since there are four children, and each child has two possible outcomes (B or G), the total number of combinations is 2^4 = 16. The sample space includes all permutations of B and G, such as {BBBB, BBGG, BGGB, etc.}. Write out all 16 combinations explicitly.
3. Step 3: Identify the favorable outcomes. To find the probability of getting three girls and one boy, look for all arrangements in the sample space where there are exactly three G's and one B. For example, {GGGB, GGBG, GBGG, BGGG}. Count the number of such favorable outcomes.
4. Step 4: Calculate the probability. The probability of an event is given by the formula P(Event) = (Number of favorable outcomes) / (Total number of outcomes in the sample space). Use the count of favorable outcomes from Step 3 and the total number of outcomes (16) to set up the probability calculation.
5. Step 5: Simplify the probability expression. If needed, simplify the fraction obtained in Step 4 to its lowest terms to express the probability in its simplest form.

Key Concepts

Sample Space

The sample space is the set of all possible outcomes of a random experiment. In the context of having four children, the sample space includes all combinations of boys and girls, represented as sequences like 'GGGB', 'GBGG', etc. Understanding the sample space is crucial for calculating probabilities, as it provides the foundation for determining how many favorable outcomes exist.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. To find the probability of getting three girls and one boy in any order from the sample space of four children, you would count the number of favorable outcomes (combinations of three girls and one boy) and divide it by the total number of outcomes in the sample space.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In this scenario, it helps determine how many different ways three girls and one boy can be arranged among four children. The formula for combinations can be used to calculate the number of ways to choose positions for the girls, which is essential for finding the probability of the desired outcome.