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Chapter 5: Problem 16

In Problems 13-18, find the probability of the indicated event if \(P(E)=0.25\) and \(P(F)=0.45\) \(P(E\) and \(F)\) if \(E\) and \(F\) are mutually exclusive

Short Answer

Expert verified
Since E and F are mutually exclusive, \(P(E \text{ and } F) = 0\).

Step by step solution

01

Understand Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. This means if event E happens, event F cannot happen, and vice versa.
02

Definition of Probability for Mutually Exclusive Events

For mutually exclusive events, the probability of both events occurring together is zero. Therefore, if E and F are mutually exclusive, the probability of both E and F occurring, denoted as P(E and F), is 0.
03

Apply the Information

Since it is given that E and F are mutually exclusive, directly apply the fact that their joint probability is zero: \(P(E \text{ and } F) = 0\)

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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 branch of mathematics that deals with the likelihood or chance of different outcomes. It's a way to mathematically quantify uncertainty and risk. We use probability to predict the chances of different events happening in situations with random variability. A probability value ranges between 0 and 1, where 0 means the event will not happen and 1 means the event is certain to happen.

There are a few key concepts in probability theory that are helpful to know:
  • Sample Space: The set of all possible outcomes of a random process.
  • Event: A subset of the sample space; it is the specific outcome or group of outcomes we are interested in.
  • Probability of an Event: The measure of the chance that the event will occur, usually denoted as P(E).
  • Complementary Events: If E is an event, then the complement of E, denoted as E', includes all outcomes not in E. The probabilities of E and E' add up to 1.
events in probability
Events in probability often involve calculating the likelihood of different types of outcomes. An event can comprise one or several outcomes from the sample space. Understanding the nature of events and how they interact is crucial for solving probability problems.

Here are some important types of events:
  • Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.
  • Dependent Events: The occurrence of one event does impact the likelihood of another event happening.
  • Mutually Exclusive Events: These are events that cannot happen at the same time. For instance, if E and F are mutually exclusive, then \(P(E \text{ and } F) = 0\).

In mutually exclusive events, the probability that either event E or F occurs is given by:

\(P(E \text{ or } F) = P(E) + P(F)\).

This sum is useful in various probability scenarios and helps to calculate compound probabilities.
joint probability
Joint probability refers to the likelihood of two events happening at the same time. It鈥檚 denoted as \(P(E \text{ and } F)\) where E and F are the events involved. This can be a crucial aspect in complex problems where multiple events are combined.

For two events E and F:
  • If E and F are mutually exclusive, \(P(E \text{ and } F) = 0\) because they cannot occur simultaneously.
  • If E and F are not mutually exclusive, \(P(E \text{ and } F)\) can be calculated if we have additional information on how they overlap.
Understanding joint probability is essential in fields like statistics, finance, and other areas involving risk and uncertainty.

For independent events, the joint probability is the product of the probabilities of each event:

\(P(E \text{ and } F) = P(E) \times P(F)\)

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