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Chapter 5: Problem 10

Suppose events \(E\) and \(F\) are independent, \(P(E)=0.7\), and \(P(F)=0.9 .\) What is the \(P(E \text { and } F) ?\)

Short Answer

Expert verified
The probability \( P(E \text{ and } F) \) is 0.63.

Step by step solution

01

Understand the concept of independent events

Independent events are those events where the occurrence of one event does not affect the occurrence of the other. For independent events, the probability of both events happening simultaneously (intersection) is the product of their individual probabilities.
02

Write down the given probabilities

Given: \[ P(E) = 0.7 \] \[ P(F) = 0.9 \]
03

Apply the formula for independent events

Use the formula for the probability of both independent events occurring: \[ P(E \text{ and } F) = P(E) \times P(F) \]
04

Calculate the probability

Substitute the given values: \[ P(E \text{ and } F) = 0.7 \times 0.9 \] Perform the multiplication: \[ P(E \text{ and } F) = 0.63 \]

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Key Concepts

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

probability
In mathematics, probability is a way to measure the likelihood of an event happening. It ranges from 0 to 1.
  • 0 means the event will not happen.
  • 1 means the event will certainly happen.
Probability of an event E, written as \(P(E)\), can be calculated by dividing the number of ways event E can occur by the total number of possible outcomes. For example, if you roll a fair six-sided die, the probability of rolling a 4 is \(P(\text{rolling a } 4) = \frac{1}{6}\). When working with many events, such as in our exercise, it's crucial to manage the probabilities correctly to understand the combined outcomes.
intersection of events
The intersection of events refers to the scenario where two or more events occur at the same time. The symbol for the intersection is '鈭�'. For example, the intersection of events E and F is written as \(E \text{ and } F or E \bigcap F\).
For the given exercise, events E and F are independent, meaning that one event occurring does not influence the other.
This makes calculating the intersection straightforward. Using the formula for the intersection of independent events: \[P(E \text{ and } F) = P(E) \times P(F)\], you can find the probability of both E and F happening together.
multiplication rule
The multiplication rule for independent events is a fundamental concept in probability. For independent events E and F, the rule states that the probability of both events occurring is the product of their individual probabilities. Written as \[P(E \text{ and } F) = P(E) \times P(F)\].
This rule simplifies the calculation significantly.
If the probability of E is 0.7 and the probability of F is 0.9, applying the multiplication rule means: \[P(E \text{ and } F) = 0.7 \times 0.9 = 0.63\].
This is how you determine the joint probability of two independent events, making it easier to handle even more complex problems.

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Most popular questions from this chapter

The data in the following table show the results of a national study of 137,243 U.S. men that investigated the association between cigar smoking and death from cancer. Note: Current cigar smoker means "cigar smoker at time of death." $$\begin{array}{|l|c|}\hline & \text { Died from cancer } & \text { Did not die from cancer } \\ \hline \text { Never smoked cigars } & 782 & 120,747 \\\\\hline \text { Former cigar smoker } & 91 & 7,757 \\ \hline \text { Current cigar smoker } & 141 & 7,725 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected individual from the study who died from cancer was a former cigar smoker? (b) What is the probability that a randomly selected individual from the study who was a former cigar smoker died from cancer?

How many different 10 -letter words (real or imaginary) can be formed from the letters in the word STATISTICS?

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.

The notation \(P(F | E)\) means the probability of event _________ given event _________ .

The probability that a randomly selected individual in the United States 25 years and older has at least a bachelor's degree is \(0.272 .\) The probability that an individual in the United States 25 years and older has at least a bachelor's degree, given that the individual is Hispanic, is 0.114. Are the events "bachelor's degree" and "Hispanic" independent? (Source: Educational Attainment in the United States, 2003. U.S. Census Bureau, June 2004 )
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