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Chapter 5: Problem 14

Suppose that \(E\) and \(F\) are two events and \(P(E\) and \(F)=0.4\) and \(P(E)=0.9 .\) Find \(P(F \mid E)\).

Short Answer

Expert verified
P(F \, | \, E) = 0.4444

Step by step solution

01

Understand the Given Information

We are given the following probabilities: - The probability of both events E and F occurring, denoted as P(E and F) = 0.4. - The probability of event E occurring, denoted as P(E) = 0.9.
02

Identify the Formula

We need to find the conditional probability of F given E, denoted as P(F \, | \, E). The formula for conditional probability is \[ P(F \, | \, E) = \frac{P(E \, and \, F)}{P(E)} \].
03

Substitute the Given Values

Substitute P(E and F) = 0.4 and P(E) = 0.9 into the formula: \[ P(F \, | \, E) = \frac{0.4}{0.9} \].
04

Simplify the Expression

Simplify \[ P(F \, | \, E) = \frac{0.4}{0.9} \]: \[ P(F \, | \, E) = \frac{4}{9} \approx 0.4444 \].

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Key Concepts

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

Conditional Probability Formula
Conditional probability helps us find the likelihood of an event occurring, given that another event has already occurred. Symbolically, the conditional probability of event F given event E is written as \( P(F \, | \, E) \).
The formula for conditional probability is:
\[ P(F \mid E) = \frac{P(E \text{ and } F)}{P(E)} \]
This formula tells us how to compute the probability of F happening while considering that E has already taken place.
We use this formula when we have the probabilities of both events occurring together (\(P(E \text{ and } F)\)) and the probability of the condition event (\(P(E)\)).
Probability of Events
Before diving into conditional probability, we must understand the basic probabilities of individual events.
These are probabilities of single events happening without considering any other conditions.
For example, in our exercise, we first need to know:
  • \( P(E) = 0.9 \)
  • \( P(E \text{ and } F) = 0.4 \)

This means the probability of E happening is 0.9, and the probability of both E and F occurring together is 0.4.
Having these basic probabilities makes it easier to calculate more complex probabilities such as conditional probabilities.
Simplification of Fractions
Once we have the probabilistic values in our formula, we often end up with a fraction.
Simplification of these fractions is key to giving us a final, more understandable result.
Using the provided formulas and values, we had:
\( P(F \mid E) = \frac{0.4}{0.9} \)
To simplify this further, we found that:
\( \frac{0.4}{0.9} = \frac{4}{9} \)
This ratio simply means 4 out of every 9 times when E happens, F will also happen.
Simplified fractions can also help us see patterns and better understand probabilities in relative terms.
Probability Notation
Understanding how to read and write probabilities using proper notation is crucial.
It allows us to accurately communicate mathematical ideas.
For example, the notation \(P(E \mid F)\) reads as 'the probability of E given F'.
Additionally, events that happen together are written as: \( P(E \text{ and } F) \).
Each part of the notation helps convey specific information:
  • \( P(E) \) is the probability of event E.
  • \( P(E \, | \, F) \) is the probability of event E occurring given that event F has occurred.
  • \( P(E \text{ and } F) \) is the probability of both events E and F occurring simultaneously.

Being familiar with these notations ensures that we can understand and solve probability problems systematically.

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Most popular questions from this chapter

Among 21 - to 25 -year-olds, \(29 \%\) say they have driven while under the influence of alcohol. Suppose that three 21 - to 25 -year-olds are selected at random. Source: U.S. Department of Health and Human Services, reported in USA Today (a) What is the probability that all three have driven while under the influence of alcohol? (b) What is the probability that at least one has not driven while under the influence of alcohol? (c) What is the probability that none of the three has driven while under the influence of alcohol? (d) What is the probability that at least one has driven while under the influence of alcohol?

The distribution of National Honor Society members among the students at a local high school is shown in the table. A student's name is drawn at random. $$ \begin{array}{lcc} \text { Class } & \text { Total } & \text { National Honor Society } \\ \hline \text { Senior } & 92 & 37 \\ \hline \text { Junior } & 112 & 30 \\ \hline \text { Sophomore } & 125 & 20 \\ \hline \text { Freshman } & 120 & 0 \\ \hline \end{array} $$ (a) What is the probability that the student is a junior? (b) What is the probability that the student is a senior, given that the student is in the National Honor Society?

A woman has five blouses and three skirts. Assuming that they all match, how many different outfits can she wear?

In finance, a derivative is a financial asset whose value is determined (derived) from a bundle of various assets, such as mortgages. Suppose a randomly selected mortgage has a probability of 0.01 of default. (a) What is the probability a randomly selected mortgage will not default (that is, pay off)? (b) What is the probability a bundle of five randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others? Note: A derivative might be an investment in which all five mortgages do not default. (c) What is the probability the derivative becomes worthless? That is, at least one of the mortgages defaults? (d) In part (b), we made the assumption that the likelihood of default is independent. Do you believe this is a reasonable assumption? Explain.

According to the U.S. Census Bureau, the probability that a randomly selected household speaks only English at home is \(0.81 .\) The probability that a randomly selected household speaks only Spanish at home is 0.12 . (a) What is the probability that a randomly selected household speaks only English or only Spanish at home? (b) What is the probability that a randomly selected household speaks a language other than only English or only Spanish at home? (c) What is the probability that a randomly selected household speaks a language other than only English at home? (d) Can the probability that a randomly selected household speaks only Polish at home equal 0.08 ? Why or why not?
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