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Chapter 5: Problem 19

A couple plans to have three children. Suppose that the probability of any given child being female is \(0.5,\) and suppose that the genders of each child are independent events. a. Write out all outcomes in the sample space for the genders of the three children. b. What should be the probability associated with each outcome? Using the sample space constructed in part a, find the probability that the couple will have c. two girls and one boy. d. at least one child of each gender.

Short Answer

Expert verified
a. Sample Space: {GGG, GGB, GBG, BGG, BBG, BGB, GBB, BBB}. b. Probability per outcome: 0.125. c. Two girls and one boy: 0.375. d. At least one of each gender: 0.75.

Step by step solution

01

Identify Sample Space

For each of the three children, there are two possible outcomes: girl (G) or boy (B). Therefore, we can list the possible combinations: GGG, GGB, GBG, BGG, BBG, BGB, GBB, BBB.
02

Calculate Total Outcomes

Since there are two possibilities per child and three children, the total combinations can be calculated as \(2^3 = 8\). So, there are 8 possible outcomes in the sample space.
03

Probability of Each Outcome

Each child being a girl or boy is independent with a probability of 0.5. Therefore, the probability of any specific sequence is \((0.5)^3 = 0.125\). All outcomes are equally likely.
04

Calculate Probability for Two Girls and One Boy

The configurations for two girls and one boy are: GGB, GBG, BGG. There are 3 such sequences. Therefore, the probability is \(3 \times 0.125 = 0.375\).
05

Calculate Probability for At Least One Child of Each Gender

Outcomes with at least one child of each gender are all except GGG and BBB: GGB, GBG, BGG, BBG, BGB, GBB. There are 6 such sequences. Therefore, the probability is \(6 \times 0.125 = 0.75\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Space
In probability theory, the sample space refers to the set of all possible outcomes of a particular experiment or event. For the scenario involving a couple planning to have three children, each child can either be a girl (G) or a boy (B). Therefore, we construct the sample space by considering all the possible combinations of girl and boy for three children.

Here's how it works:
  • With three consecutive children, each with two possible outcomes (G or B), the sample space is represented as a list of all potential sequences.
  • The possible outcomes are GGG, GGB, GBG, BGG, BBG, BGB, GBB, and BBB.
Understanding the sample space helps us predict and calculate probabilities for specific gender combinations in the future children's set.
Independent Events
In this context, independent events are a crucial part of understanding probability. An independent event means that the outcome of one event does not affect the outcome of another event. In the exercise about the couple having three children, each child's gender is considered to be independent of the others.

Why does this matter?
  • It allows us to calculate the probability of combinations since the likelihood of one child's gender doesn't change because of another child's gender.
  • Each child has a probability of 0.5 to be a girl or a boy, independent of what the other children's genders are.
This makes it simple to multiply probabilities for each child separately when calculating combined probabilities for gender sequences.
Combinatorics
Combinatorics deals with counting, grouping, and arranging different sets of elements. It's a key concept in determining possible sequences in probability exercises like this one about predicting children's genders.

In this particular exercise:
  • We use combinatorics to list all the possible gender sequences for the three children.
  • Given two possible genders for each child, there are a total of eight combinations (because the calculation is done via the formula \(2^n\), where \(n\) is the number of children).
Combinatorics helps simplify complex problems because it provides us with methods to thoroughly count and consider all possible outcomes.
Outcome Probability
The outcome probability is the likelihood that a particular event or sequence will occur. Since we have established the sample space of gender sequences for the three children and identified the independent nature of each event, assigning probabilities to each outcome becomes clearer.

Here's how we calculate it:
  • Each gender sequence has a probability of occurring which is based on the factor that each child's gender is independent. Therefore, each sequence has a probability of \((0.5)^3 = 0.125\).
  • The probability for a specific event, such as having two girls and one boy, is calculated by finding the sequences that match the criteria (GGB, GBG, BGG) and multiplying their individual probabilities.
This systematic approach allows us to compute the chances of different outcomes, allowing predictions and strategic decisions based on the computed probabilities.

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Most popular questions from this chapter

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