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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.2.35
Chapter 3, Problem 3.2.35

"According to Bayes’ Theorem, the probability of event A , given that event B has occurred, is
P(A|B) = P(A) * P(B|A)P(A) * P(B|A) + P(A') * P(B|A').
In Exercises 33–38, use Bayes’ Theorem to find P(A|B).
35. P(A) = 0.25, P(A') = 0.75, P(B|A) = 0.3 , and P(B|A') = 0.5 "

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1
Step 1: Recall Bayes' Theorem formula: P(A|B) = (P(A) * P(B|A)) / (P(A) * P(B|A) + P(A') * P(B|A')). This formula helps calculate the conditional probability of event A given that event B has occurred.
Step 2: Identify the given values from the problem: P(A) = 0.25, P(A') = 0.75, P(B|A) = 0.3, and P(B|A') = 0.5.
Step 3: Substitute the given values into the numerator of the formula: P(A) * P(B|A) = 0.25 * 0.3.
Step 4: Substitute the given values into the denominator of the formula: P(A) * P(B|A) + P(A') * P(B|A') = (0.25 * 0.3) + (0.75 * 0.5).
Step 5: Simplify the numerator and denominator expressions separately, then divide the numerator by the denominator to find P(A|B).

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Key Concepts

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

Bayes' Theorem

Bayes' Theorem is a fundamental principle in probability theory that describes how to update the probability of a hypothesis based on new evidence. It states that the probability of event A given event B (P(A|B)) can be calculated using the formula P(A|B) = P(A) * P(B|A) / (P(A) * P(B|A) + P(A') * P(B|A')). This theorem is particularly useful in scenarios where prior knowledge about the events is available.
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Conditional Probability

Conditional probability is the measure of the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which represents the probability of event A occurring under the condition that event B is true. Understanding conditional probability is crucial for applying Bayes' Theorem, as it allows us to assess how the occurrence of one event influences the likelihood of another.
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Prior and Posterior Probabilities

In the context of Bayes' Theorem, prior probability refers to the initial assessment of the likelihood of an event before new evidence is considered (P(A)), while posterior probability is the updated probability after taking into account the new evidence (P(A|B)). The distinction between these two types of probabilities is essential for understanding how new information can change our beliefs about the likelihood of events.
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Related Practice
Textbook Question

1. When two events are mutually exclusive, why is P(A and B) = 0?

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Textbook Question

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

3. A combination is an ordered arrangement of objects.

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Textbook Question

Cards In Exercises 59-62, you are dealt a hand of five cards from a standard deck of 52 playing cards.

62. Find the probability of being dealt three of a kind (the other two cards are different from each other).

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Textbook Question

Matching Probabilities In Exercises 11-16, match the event with its probability.

a. 0.95

b. 0.005

c. 0.25

d. 0

e. 0.375

f. 0.5

11. A random number generator is used to select a number from 1 to 100. What is the probability of selecting the number 153?

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Textbook Question

Classifying Events Based on Studies In Exercises 15-18, identify the two events described in the study. Do the results indicate that the events are independent or dependent? Explain your reasoning.

17. A study found that there is no relationship between playing violent video games and aggressive or bullying behavior in teenagers. (Source: The Royal Society Publishing)

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Textbook Question

Cards In Exercises 59-62, you are dealt a hand of five cards from a standard deck of 52 playing cards.

59. Find the probability of being dealt two clubs and one of each of the other three suits.

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