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Chapter 5: Problem 15

Find the probability of the indicated event if \(P(E)=0.25\) and \(P(F)=0.45\) Find \(P(E \text { or } F)\) if \(E\) and \(F\) are mutually exclusive.

Short Answer

Expert verified
The probability is 0.70.

Step by step solution

01

Understand the problem

Identify that the problem is about calculating the probability of either event E or event F happening, given that they are mutually exclusive.
02

Definition of Mutually Exclusive Events

Recall that mutually exclusive events are events that cannot occur at the same time. Thus, for mutually exclusive events, the probability of both events occurring together is zero, i.e., \[P(E \text{ and } F) = 0\]
03

Use the Addition Rule

For mutually exclusive events, the probability that either event E or event F occurs is the sum of their individual probabilities. This is given by: \[P(E \text{ or } F) = P(E) + P(F)\]
04

Substitute the Values

Substitute the given probabilities into the addition rule equation: \[P(E \text{ or } F) = 0.25 + 0.45\]
05

Calculate the Result

Perform the addition: \[0.25 + 0.45 = 0.70\]
06

State the Final Answer

Therefore, the probability that either event E or event F occurs is 0.70.

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Key Concepts

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

Mutually Exclusive Events
In probability theory, two events are considered *mutually exclusive* if they cannot happen at the same time. This means that the occurrence of one event precludes the occurrence of the other. For example:
  • Flipping a coin and getting heads or tails - these events are mutually exclusive because the coin can only show one side at a time.
  • Drawing a card from a standard deck and getting an Ace or a King - these events are mutually exclusive because you cannot draw a card that is both an Ace and a King simultaneously.
In mathematical terms, if Events E and F are mutually exclusive, then the probability of both events occurring together is zero: \[ P(E \text{ and } F) = 0 \]. Understanding whether events are mutually exclusive is crucial because it directly affects how we calculate the probability of their union.
Addition Rule in Probability
The *Addition Rule in Probability* helps us find the probability of either one of two events happening. For mutually exclusive events, where the two events cannot happen simultaneously, the addition rule simplifies to:
  • \[ P(E \text{ or } F) = P(E) + P(F) \]
This means we simply add the probabilities of each event. Let's take an example:
Suppose we have Event E with a probability of 0.25 and Event F with a probability of 0.45. If E and F are mutually exclusive, the probability that either E or F occurs is: \[ P(E \text{ or } F) = 0.25 + 0.45 = 0.70 \]So, the result is 0.70, meaning there is a 70% chance that either Event E or Event F will occur.
Basic Probability Concepts
Before diving into more complex problems in probability, it's crucial to understand some *basic concepts*. Here are a few key points:
  • **Probability** - It is a measure of how likely an event is to occur, represented by a number between 0 and 1. A probability of 0 means the event cannot happen, while a probability of 1 means it is certain to happen.
  • **Event** - This is any outcome or group of outcomes from a random experiment. For instance, rolling a die and getting a 3 is an event.
  • **Sample Space** - This is the set of all possible outcomes. For a die, the sample space is {1, 2, 3, 4, 5, 6}.
  • **Mutually Exclusive Events** - As discussed, these are events that cannot occur at the same time.
  • **Addition Rule** - Used to find the probability of either of two events occurring.
These foundational concepts will help you understand and solve a wide range of probability problems more effectively. Always identify whether events are mutually exclusive, as this greatly influences how you apply probability rules.

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Most popular questions from this chapter

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