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Chapter 4: Problem 78

Given that \(P(A \mid B)=.40\) and \(P(A\) and \(B)=.36\), find \(P(B)\).

Short Answer

Expert verified
The probability of B, denoted as \(P(B)\), is .90 or 90%.

Step by step solution

01

Write down the formula for conditional probability

The formula for conditional probability is: \(P(A \mid B) = \frac{P(A \; and \; B)}{P(B)}\)
02

Plug in the given values

We know that \(P(A \mid B)=.40\) and \(P(A \; and \; B)=.36\), so we plug in these values: \(.40 = \frac{.36}{P(B)}\)
03

Solve for P(B)

To find the probability of B, we need to isolate \(P(B)\). We do this by multiplying both sides of the equation by \(P(B)\), which leaves us with \(.40 \cdot P(B) = .36\). Then, divide both sides by .40 to solve for \(P(B)\). This gives us \(P(B) = \frac{.36}{.40}\)

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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 foundation of understanding how likely events are to occur. It helps answer questions about the likelihood of various outcomes. In probability terms, we often talk about events and their probabilities. An event is any outcome or combination of outcomes. Imagine rolling a six-sided die. Each number on the die represents a possible event.
The two basic rules are:
  • The probability of an event always ranges from 0 to 1, where 0 means the event will not occur, and 1 means it will occur.
  • The sum of the probabilities of all possible outcomes is 1.
These basic principles help us to calculate and understand more complex probabilities, like in our exercise.
Event Dependence
Event dependence refers to how the occurrence of one event affects the probability of another. This concept is crucial in understanding conditional probability. Two events are independent if the occurrence of one does not change the probability of the other. For instance, flipping a coin twice: the result of the first flip does not influence the second.
However, if events are dependent, as in our exercise, the probability of one event changes based on another. Conditional probability helps us describe this interaction.
  • Imagine a deck of cards: pulling an ace is a different probability if you鈥檝e already pulled a card.
  • Understanding dependence helps solve problems like finding the conditional probability, where one event has already occurred.
This concept helps in practical scenarios, like predicting weather or assessing risk.
Formula Derivation
Deriving formulas helps us solve probability problems step-by-step. For conditional probability, the formula is:\[ P(A \mid B) = \frac{P(A \; \text{and} \; B)}{P(B)} \]This formula helps find the probability of an event A occurring, given that event B has occurred. In the provided exercise, we use this formula to find \(P(B)\).
To derive \(P(B)\):
  • First, identify the given values: \(P(A \mid B) = 0.40\) and \(P(A \; \text{and} \; B) = 0.36\).
  • Plug these into the formula: \[ 0.40 = \frac{0.36}{P(B)} \]
  • Multiply both sides by \(P(B)\) to isolate it.\[ 0.40 \cdot P(B) = 0.36 \]
  • Finally, divide by 0.40 to find \(P(B)\):\[ P(B) = \frac{0.36}{0.40} = 0.90 \]
This step-by-step breakdown makes deriving and solving probability problems more digestible.

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Most popular questions from this chapter

According to a 2007 America's Families and Living Arrangements Census Bureau survey, \(52.1\) million children lived with both of their parents in the same household, whereas \(21.6\) million lived with at most one parent in the household. Assume that all U.S. children are included in this survey and that this information is true for the current population. If one child is selected at random, what are the two complementary events and their probabilities?

A contractor has submitted bids for two state construction projects. The probability of winning each contract is \(.25\), and it is the same for both contracts a. What is the probability that he will win both contracts? b. What is the probability that he will win neither contract? Draw a tree diagram for this problem.

A pizza parlor has 12 different toppings available for its pizzas, and 2 of these toppings are pepperoni and anchovies. If a customer picks 2 toppings at random, find the probability that a. neither topping is anchovies b. pepperoni is one of the toppings

The probability that a family owns a washing machine is \(.68\), that it owns a DVD player is 81 , and that it owns both a washing machine and a DVD player is \(.58\). What is the probability that a randomly selected family owns a washing machine or a DVD player?

In a class of 35 students, 13 are seniors, 9 are juniors, 8 are sophomores, and 5 are freshmen. If one student is selected at random from this class, what is the probability that this student is a. a junior? b. a freshman?
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