Question

For the experiments, list the simple events in the sample space, assign probabilities to the simple events, and find the required probabilities. Three children are selected, and their gender recorded. Assume that males and females are equally likely. What is the probability that there are two boys and one girl in the group?

Short answer

Answer: The probability of having two boys and one girl in the group is 3/8.

Step-by-step solution

List the sample space

There are 2^3 = 8 possible gender arrangements for three children (BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG).

Assign probabilities to the simple events

Since the genders are equally likely, for each child, there is a probability of 0.5 of being a boy and 0.5 of being a girl. We multiply the probabilities of each child's gender for each arrangement in the sample space: BBB = (0.5)(0.5)(0.5) = 1/8 BBG = (0.5)(0.5)(0.5) = 1/8 BGB = (0.5)(0.5)(0.5) = 1/8 GBB = (0.5)(0.5)(0.5) = 1/8 GGB = (0.5)(0.5)(0.5) = 1/8 GBG = (0.5)(0.5)(0.5) = 1/8 BGG = (0.5)(0.5)(0.5) = 1/8 GGG = (0.5)(0.5)(0.5) = 1/8

Find the required probability

There are three arrangements with two boys and one girl: BBG, BGB, and GBB. We sum their probabilities: P(two boys and one girl) = P(BBG) + P(BGB) + P(GBB) = 1/8 + 1/8 + 1/8 = 3/8. Hence, the probability of having two boys and one girl in the group is 3/8.

Key concepts

Sample Space

In the realm of probability, the concept of sample space is crucial. It represents the set of all possible outcomes of a given experiment. In the context of our example, the experiment involves recording the genders of three children.
For three children, each having a gender of either male or female, you can think of the sample space as all the different ways their genders can be recorded.
This includes every possible combination, such as boys and girls in different orders.
If we denote a boy by 'B' and a girl by 'G', the sample space can be written as follows:

• BBB
• BBG
• BGB
• GBB
• GGB
• GBG
• BGG
• GGG

Each sequence represents a unique arrangement of three genders. The sample space consists of 2 multiplied by itself three times, written as 2^3, resulting in 8 possible outcomes. So, these eight sequences make up the entire sample space for this scenario.

Simple Events

Simple events are specific outcomes within the sample space. They are the individual results that can occur from the experiment. In our example with three children, simple events are each unique arrangement of boys and girls we enumerated earlier.
Each sequence such as 'BBB', 'BBG', 'BGB', etc., represents a simple event. For instance, 'BBG' is one simple event, which would mean the first two children are boys and the last child is a girl.
Understanding the nature of simple events helps us identify the likelihood of each specific outcome happening. In this example, there are eight equally probable simple events, thanks to the assumption that each child is equally likely to be a boy or a girl.

Equally Likely Outcomes

The term "equally likely outcomes" refers to scenarios where each possible outcome has the same probability of occurring. This is essential for calculating probabilities in a uniform way over the sample space.
In our example, each gender outcome, such as a child being a boy or a girl, has a probability of 0.5. When we expand this to record the gender of three children, each arrangement or simple event—like 'BBB' or 'GGB'—is equally likely. Each one of these eight arrangements occurs with the same probability.
Ultimately, when events are equally likely, it simplifies the calculation of the probability for any specific arrangement by utilizing the basic principle: \[ P(Event) = \frac{1}{\text{Number of Outcomes}} \] Therefore, for each simple event in our situation, the probability is 1/8.

Gender Probability

In probability theory, calculating the probability of different gender combinations provides insight into the likelihood of varying gender distributions. This particular problem involves determining the probability of getting two boys and one girl out of three children.
To find this, we first identify the favorable outcomes from the sample space. In our list, 'BBG', 'BGB', and 'GBB' are the sequences that fit the criteria of having two boys and one girl.
Each sequence has an individual probability of 1/8, as established due to equally likely outcomes. The calculation for the desired probability involves summing the probabilities of these sequences:

• \( P(\text{two boys and one girl}) = P(BBG) + P(BGB) + P(GBB) \)
• \( P(\text{two boys and one girl}) = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8} \)

Thus, the probability that two of the children are boys and one is a girl is 3/8, demonstrating how gender probabilities can be computed using simple event probabilities.