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Chapter 5: Problem 13

Find the probability of the indicated event if \(P(E)=0.25\) and \(P(F)=0.45\) Find \(P(E \text { or } F)\) if \(P(E \text { and } F)=0.15\)

Short Answer

Expert verified
The probability is 0.55.

Step by step solution

01

Understand the Question

We are given the probabilities of two events, E and F, i.e., \(P(E) = 0.25\) and \(P(F) = 0.45\). We are also given the probability of both events happening together, i.e., \(P(E \text{ and } F) = 0.15\). The goal is to find \(P(E \text{ or } F)\).
02

Apply the Formula for Union of Events

The formula to find the probability of the union of two events is given by: \[P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)\].
03

Substitute the Given Values

Substitute the given probabilities into the formula: \[P(E \text{ or } F) = 0.25 + 0.45 - 0.15\].
04

Simplify the Expression

Perform the arithmetic operations: \[P(E \text{ or } F) = 0.25 + 0.45 - 0.15 = 0.55\].

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Key Concepts

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

probability of events
Probability helps us measure the likelihood of an event occurring. It's represented by a number between 0 and 1. An event with a probability of 0 means it will never happen, while an event with a probability of 1 means it will always happen. Most events fall somewhere in between. In our example, we are given the probabilities of two events, E and F: \(P(E) = 0.25\) and \(P(F) = 0.45\). This tells us that event E has a 25% chance of occurring and event F has a 45% chance of occurring.
union of events
The union of two events refers to the probability that either event E, event F, or both events occur. We write this as \(P(E \text{ or } F)\). To find this probability, we use the formula: \[ P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) \] The subtraction of \(P(E \text{ and } F)\) is necessary because if we simply add \(P(E)\) and \(P(F)\), we would count the probability of both events happening together twice. By subtracting \(P(E \text{ and } F)\), we correct this.
intersection of events
The intersection of two events is the probability that both events occur at the same time. We write this as \(P(E \text{ and } F)\). In our example, this probability is given as \(P(E \text{ and } F) = 0.15\). This means there is a 15% chance that both event E and event F will happen simultaneously. This value is crucial for calculating the union of two events, as seen in the previous section.
arithmetic operations in probability
Arithmetic operations are fundamental in probability calculations. To find the union of events, we perform addition and subtraction: \[ P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) \]
Let's see how the arithmetic works out with the given values:
  • First, add the probabilities of E and F: \(0.25 + 0.45 = 0.70\).
  • Then, subtract the intersection: \(0.70 - 0.15 = 0.55\).
Therefore, \(P(E \text{ or } F) = 0.55\). This step-by-step approach makes it easier to understand how to use arithmetic in probability to solve problems.

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Most popular questions from this chapter

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