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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

John, Roberto, Clarice, Dominique, and Marco work for a publishing company. The company wants to send two employees to a statistics conference in Orlando. To be fair, the company decides that the two individuals who get to attend will have their names drawn from a hat. This is like obtaining a simple random sample of size 2 (a) Determine the sample space of the experiment. That is, list all possible simple random samples of size \(n=2\) (b) What is the probability that Clarice and Dominique attend the conference? (c) What is the probability that Clarice attends the conference? (d) What is the probability that John stays home?

Exclude leap years from the following calculations and assume each birthday is equally likely: (a) Determine the probability that a randomly selected person has a birthday on the 1 st day of a month. Interpret this probability. (b) Determine the probability that a randomly selected person has a birthday on the 31 st day of a month. Interpret this probability. (c) Determine the probability that a randomly selected person was born in December. Interpret this probability. (d) Determine the probability that a randomly selected person has a birthday on November \(8 .\) Interpret this probability. (e) If you just met somebody and she asked you to guess her birthday, are you likely to be correct? (f) Do you think it is appropriate to use the methods of classical probability to compute the probability that a person is born in December?

How many different simple random samples of size 5 can be obtained from a population whose size is \(50 ?\)

Suppose you are dealt 5 cards from a standard 52 -card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52 card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4 th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

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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