Ch. 4 - Probability

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 4 - Probability

Problem 4.3.6

Chapter 4, Problem 4.3.6

Probability of a Girl Assuming that boys and girls are equally likely, find the probability of a couple having a boy when their third child is born, given that the first two children were both girls.

Verified step by step guidance

Step 1: Understand the problem. The question asks for the probability of the third child being a boy, given that the first two children are girls. Note that the genders of the children are independent events, meaning the outcome of one child does not affect the outcome of another.

Step 2: Recall the basic probability rule. When boys and girls are equally likely, the probability of having a boy or a girl for any single child is 1/2 (or 0.5).

Step 3: Recognize that the condition provided (the first two children being girls) does not influence the probability of the third child’s gender. This is because the events are independent.

Step 4: Write the probability of the third child being a boy as P(Boy) = 1/2. This is based solely on the fact that boys and girls are equally likely, and the previous outcomes do not affect this probability.

Step 5: Conclude that the probability of the third child being a boy remains 1/2, regardless of the genders of the first two children. This is a key concept in understanding independent events in probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In this scenario, we are interested in the probability of having a boy as the third child, given that the first two children were girls. This concept is crucial for understanding how prior outcomes can influence the probability of future events.

Independence of Events

In probability, two events are considered independent if the occurrence of one does not affect the occurrence of the other. In this case, the gender of the third child is independent of the genders of the first two children. This means that regardless of the first two being girls, the probability of the third child being a boy remains unchanged.

Sample Space

The sample space is the set of all possible outcomes of a probabilistic experiment. For the birth of children, the sample space consists of combinations of boys and girls. In this problem, the relevant outcomes for the third child are 'boy' or 'girl', and understanding the sample space helps clarify the probabilities associated with each outcome.

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Related Practice

Textbook Question

In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Texting and Alcohol If one of the high school drivers is randomly selected, find the probability that the selected driver did not text while driving and did not drive when drinking.

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Textbook Question

In Exercises 33–40, use the given probability value to determine whether the sample results are significant.

Voting Repeat the preceding Exercise 33 after replacing 40 Democrats being placed on the first line of voting ballots with 26 Democrats being placed on the first line. The probability of getting a result as high as 26 is 0.058638.

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Textbook Question

In Exercises 17–20, refer to the accompanying table showing results from experiments conducted by researchers Charles R. Honts (Boise State University) and Gordon H. Barland (Department of Defense Polygraph Institute). In each case, it was known whether or not the subject lied, so the table indicates when the polygraph (lie detector) test was correct.

False Negative Find the probability of selecting a subject with a negative polygraph result, given that the subject lied. What would be an unfavorable consequence of this error?

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Textbook Question

Exclusive Or The exclusive or means either one or the other event occurs, but not both.

If one of the high school drivers is randomly selected, find the probability of getting one who texted while driving or drove when drinking alcohol.

b. Repeat Exercise 11 “Texting or Drinking” using the exclusive or instead of the inclusive or.

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Textbook Question

Women in Movies In a recent year, speaking characters in movies were 68.2% male. What is the probability of randomly selecting a character with a speaking part and getting a female? What should be the value of that probability?

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Textbook Question

Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.

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