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Chapter 4: Problem 51

Given that \(P(A \mid B)=.44\) and \(P(A\) and \(B)=.33\), find \(P(B)\).

Short Answer

Expert verified
The probability of event B occurring is \(0.75\).

Step by step solution

01

Understanding the formula

First, we need to understand the formula of conditional probability which is given by: \(P(A and B) = P(A | B) * P(B)\). This formula expresses the relationship between the probability of events A and B happening together and the probability of event A happening given that B has happened.
02

Rearranging the formula

From the formula above, we can express the probability of event B happening in terms of the other known quantities: \(P(B) = P(A and B) / P(A | B)\). The solution comes straight from this formula.
03

Substituting the values into the formula

Now we substitute the given values into the formula: \(P(B) = .33 / .44\). If we divide these two values, we get the probability of event B occurring.

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Key Concepts

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

Probability Theory
Probability theory is the mathematical framework that allows us to reason about uncertainty and randomness. It helps in predicting how likely events are to happen. This field uses numbers between 0 and 1 to express the probability of an event, with 0 indicating it鈥檚 impossible and 1 meaning it鈥檚 certain.

Here's a quick overview:
  • Probability can describe a single event occurring, such as rolling a die and getting a six, or multiple events, like rolling two dice and both showing sixes.
  • The total probability of all possible outcomes of a random process always equals 1.
  • Cross connections exist between events, often described by concepts like independent, mutually exclusive, or conditional probabilities.
Understanding these basics can help us tackle various types of problems, whether simple or complex.
Bayes' Theorem
Bayes' Theorem is a powerful concept in probability theory. It provides a way to update our beliefs about the likelihood of an event based on new evidence. This theorem is particularly useful when we deal with conditional probabilities.

Bayes' Theorem states that:\[P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\]This equation helps us understand how the probability of event A occurring changes if we know that event B has occurred. Typically, it relates "new" and "prior" information.
  • P(A | B): Probability of A given B.
  • P(B | A): Probability of B given A.
  • P(A): Overall probability of A occurring.
  • P(B): Overall probability of B occurring.
Using Bayes' Theorem, we can approach problems from a perspective of evidence keeping, where data updates our understanding of events.
Probability of Events
The probability of events refers to the chance that particular outcomes will occur in a random experiment. We often deal with combinations of events, which is where joint and conditional probabilities come into play.

Some important points about the probability of events include:
  • Joint Probability: This is the probability that two events happen at the same time, expressed as \(P(A \text{ and } B)\).
  • Conditional Probability: This measures how likely an event A is to happen given that another event B has already occurred. It's written as \(P(A | B)\).
  • The formula \(P(A \text{ and } B) = P(A | B) \cdot P(B)\) helps in finding the occurrence of event B using known quantities.
Understanding these concepts facilitates breaking down problems into manageable parts, like solving for unknown probabilities as seen in many textbook exercises.

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Most popular questions from this chapter

Two students are randomly selected from a statistics class, and it is observed whether or not they suffer from math anxiety. List all the outcomes included in each of the following events. Indicate which are simple and which are compound events. a. Both students suffer from math anxiety. b. Exactly one student suffers from math anxiety. c. The first student does not suffer and the second suffers from math anxiety. d. None of the students suffers from math anxiety.

How many different outcomes are possible for four rolls of a die?

An automated teller machine at a local bank is stocked with \(\$ 10\) and \(\$ 20\) bills. When a customer withdraws \(\$ 40\) from this machine, it dispenses either two \(\$ 20\) bills or four \(\$ 10\) bills. Two customers withdraw \(\$ 40\) each. List all of the outcomes in each of the following events and mention which of these are simple and which are compound events. a. Exactly one customer receives \(\$ 20\) bills. b. Both customers receive \(\$ 10\) bills. c. At most one customer receives \(\$ 20\) bills. d. The first customer receives \(\$ 10\) bills and the second receives \(\$ 20\) bills.

The probability that an employee at a company is a female is .36. The probability that an employee is a female and married is .19. Find the conditional probability that a randomly selected employee from this company is married given that she is a female.

Consider the following addition rule to find the probability of the union of two events \(A\) and \(B\) : $$ P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B) $$ When and why is the term \(P(A\) and \(B\) ) subtracted from the sum of \(P(A)\) and \(P(B) ?\) Give one example where you might use this formula.
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