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Chapter 5: Problem 2

Suppose that \(E\) and \(F\) are two events and that \(P(E \text { and } F)=0.6\) and \(P(E)=0.8 .\) What is \(P(F | E) ?\)

Short Answer

Expert verified
0.75

Step by step solution

01

- Understand Given Information

Identify the given values: \(P(E \text{ and } F) = 0.6\) and \(P(E) = 0.8\)
02

- Recall the Formula for Conditional Probability

The formula for conditional probability is \(P(F | E) = \frac{P(E \text{ and } F)}{P(E)}\)
03

- Substitute the Given Values into the Formula

Substitute \(P(E \text{ and } F) = 0.6\) and \(P(E) = 0.8\) into the formula: \[P(F | E) = \frac{0.6}{0.8}\]
04

- Calculate the Result

Perform the division to find \(P(F | E)\): \[P(F | E) = 0.75\]

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Key Concepts

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

probability
Probability is a measure of the likelihood that a particular event will occur. It ranges from 0 (the event will not occur) to 1 (the event is certain to occur). For example, when tossing a fair coin, the probability of landing on heads is 0.5, since there are two equally likely outcomes.
When we talk about events in probability, we're often interested in events that occur in combination, like getting two heads when tossing two coins. This is known as joint probability and is represented as \( P(A \text{ and } B) \).
For the given exercise, the joint probability is 0.6, which means there's a 60% chance for both events \(E\) and \(F\) to occur together.
conditional probability formula
Conditional probability is the probability of an event occurring given that another event has already occurred. It's denoted as \( P(A | B) \), which means the probability of event \( A \) happening given that event \( B \) has happened.
The formula for conditional probability is: \[ P(F | E) = \frac{P(E \text{ and } F)}{P(E)} \]
This formula helps us to calculate probabilities under specific conditions. In our exercise, we want to find \( P(F | E) \), meaning the probability of event \(F\) occurring given that event \(E\) is true. We use the values \( P(E \text{ and } F) = 0.6 \) and \( P(E) = 0.8 \).
calculating probabilities
To calculate \( P(F | E) \) using the conditional probability formula, follow these steps:
  • First, identify the given values: \( P(E \text{ and } F) = 0.6 \) and \( P(E) = 0.8 \).
  • Next, substitute these values into the formula: \[ P(F | E) = \frac{P(E \text{ and } F)}{P(E)} \]
  • Then, perform the division: \[ \frac{0.6}{0.8} = 0.75 \]

The result tells us that given event \( E\) has occurred, there is a 75% chance that event \( F\) will occur. This calculation is essential in understanding how the occurrence of one event affects the probability of another event.

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Most popular questions from this chapter

The probability that a randomly selected 40-year-old female will live to be 41 years old is 0.99856 according to the National Vital Statistics Report, Vol. 48, No. 18 (a) What is the probability that two randomly selected \(40-\) year-old females will live to be 41 years old? (b) What is the probability that five randomly selected \(40-\) year-old females will live to be 41 years old? (c) What is the probability that at least one of five randomly selected 40 -year-old females will not live to be 41 years old? Would it be unusual that at least one of five randomly selected 40 -year-old females will not live to be 41 years old?

In a recent basketball game, a player who makes \(65 \%\) of his free throws made eight consecutive free throws. Assuming free-throw shots are independent, determine whether this feat was unusual.

Suppose a single card is selected from a standard 52 -card deck. What is the probability that the card drawn is a club? Now suppose a single card is drawn from a standard 52 -card deck, but we are told that the card is black. What is the probability that the card drawn is a club?

According to the U.S. National Center for Health Statistics, in \(2002,0.2 \%\) of deaths in the United States were 25 - to 34 -year-olds whose cause of death was cancer. In addition, \(1.97 \%\) of all those who died were 25 to 34 years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been 25 to 34 years old?

What is the probability of obtaining five heads in a row when flipping a coin? Interpret this probability.
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