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Chapter 4: Problem 5

Find the probability. Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)

Short Answer

Expert verified
The probability of having at least one girl is \( \frac{7}{8} \).

Step by step solution

01

Define the Probability

First, identify the probability of having a boy or a girl. Since boys and girls are equally likely, the probability of having a boy (B) or a girl (G) is each \(\frac{1}{2}\).
02

Total Number of Outcomes

There are three children, and each child can either be a boy or a girl. So, the total number of possible outcomes is \((2^3) = 8\) since \(2\) (boy or girl) choices for each of \(3\) children.
03

List All Possible Outcomes

List all possible outcomes: \(BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\). This shows all possible combinations of three children being boys or girls.
04

Identify Unfavorable Outcome

Identify the scenario where there are no girls: only \(BBB\) is the unfavorable outcome.
05

Calculate the Probability of Unfavorable Outcome

The probability of having no girls (only boys) is \(P(BBB) = \frac{1}{8}\).
06

Calculate the Probability of the Complement Event

The probability of having at least one girl is the complement of the probability of having no girls. Therefore, \(P(\text{at least one girl}) = 1 - P(BBB) = 1 - \frac{1}{8} = \frac{7}{8}\).

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics dealing with combinations of objects belonging to a set. In this exercise, we used combinatorics to find the total number of possible outcomes when a couple has three children. Each child can either be a boy (B) or a girl (G). Hence, we calculate the total number of outcomes by raising the number of choices (2) to the power of the number of children (3). This gives us \(2^3 = 8\) possible outcomes. Understanding combinatorics helps us systematically list all the possible combinations, which in this case are: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.
Complement Rule
The complement rule is a fundamental concept in probability. It helps us find the probability of an event by knowing the probability of its complement (the event not occurring). In this exercise, we needed to find the probability that at least one child is a girl. Instead of calculating this directly, which could be complex, we first found the probability of the complementary event鈥攈aving no girls (all boys, BBB). We knew \(P(BBB) = \frac{1}{8}\). Using the complement rule, we found the desired probability: \(P(\text{at least one girl}) = 1 - P(BBB) = 1 - \frac{1}{8} = \frac{7}{8}\).
Basic Probability
Basic probability is about measuring the likelihood of different outcomes. The probability of an event is calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this exercise, each child could either be a boy or a girl with equal probability of \(\frac{1}{2}\). With three children, there are \(2^3 = 8\) possible outcomes. By analyzing these outcomes, we found the specific probability of an event where at least one child is a girl. Basic probability tools like counting outcomes and applying simple ratios are essential to solving such problems.
Event Outcomes
Event outcomes are the possible results that can occur in a probability scenario. In the given exercise, the event outcomes are the different combinations of boys and girls a couple can have with three children. Listing all outcomes鈥擝BB, BBG, BGB, BGG, GBB, GBG, GGB, GGG鈥攈elped us see all possible scenarios. From these outcomes, we identified the single unfavorable outcome (BBB) and used it to determine the probability of having at least one girl. Understanding event outcomes enables us to methodically approach and solve probability problems effectively.

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Most popular questions from this chapter

a. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 2 of them have the same birth date. Describe the simulation and estimate the probability. b. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 3 of them have the same birth date. Describe the simulation and estimate the probability.

Express all probabilities as fractions. As of this writing, the Powerball lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 69 and, in a separate drawing, you must also select the correct single number between 1 and \(26 .\) Find the probability of winning the jackpot.

Express all probabilities as fractions. The Digital Pet Rock Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a strategic planning committee with four different members. There are 10 qualified candidates, and officers can also serve on the committee. a. How many different ways can the four officers be appointed? b. How many different ways can a committee of four be appointed? C. What is the probability of randomly selecting the committee members and getting the four youngest of the qualified candidates?

Find the indicated complements. According to the Bureau of Transportation, \(80.3 \%\) of American Airlines flights arrive on time. What is the probability of randomly selecting an American Airlines flight that does not arrive on time?

Exercise 33 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).
See all solutions

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