Problem 45 - Algebra and Trigonometry

Question

Find the probability of the indicated event if \(P(A)=0.25\) and \(P(B)=0.45\) $$P(A \cup B) \text { if } P(A \cap B)=0.15$$

Step-by-Step Solution

Answer

\( P(A \, \cup \, B) = 0.55 \)

Step 1 - Understand the Problem

We need to find the probability of the union of events A and B, denoted as \( P(A \, \cup \, B) \). We are given \( P(A) = 0.25 \), \( P(B) = 0.45 \), and \( P(A \, \cap \, B) = 0.15 \).

Step 2 - Use the Union Formula

The formula for the union of two events is \( P(A \, \cup \, B) = P(A) + P(B) - P(A \, \cap \, B) \).

Step 3 - Plug in the Values

Substitute the given values into the formula: \( P(A \, \cup \, B) = 0.25 + 0.45 - 0.15 \).

Step 4 - Simplify the Expression

Perform the arithmetic operations: \( 0.25 + 0.45 = 0.70 \) and \( 0.70 - 0.15 = 0.55 \).

Step 5 - State the Final Answer

The probability of the union of events A and B is \( P(A \, \cup \, B) = 0.55 \).

Key Concepts

Union of Events

In probability theory, the union of two events represents the occurrence of at least one of the events. If we have two events, A and B, the union of these events, denoted as \( A \, \cup \, B \), stands for all outcomes that are in either A, B, or both. Essentially, it answers the question: What is the probability that either event A happens, or event B happens, or both?

To calculate this probability, we use the formula for the union of two events:
\[ P(A \, \cup \, B) = P(A) + P(B) - P(A \, \cap \, B) \] This formula helps us avoid double-counting the outcomes that are common to both events A and B.

In our problem, we are given:
- \( P(A) = 0.25 \)
- \( P(B) = 0.45 \)
- \( P(A \, \cap \, B) = 0.15 \)

By substituting these values into the formula, we find:
\[ P(A \, \cup \, B) = 0.25 + 0.45 - 0.15 = 0.55 \] Therefore, the probability of either event A or event B occurring, or both, is 0.55.

Intersection of Events

The intersection of two events in probability theory refers to the scenarios where both events happen at the same time. If we denote two events A and B, their intersection is written as \( A \, \cap \, B \). This represents all outcomes that are in both A and B simultaneously.

For example, let's consider A and B as the events of drawing a red card and drawing a queen from a deck of cards. The intersection of A and B (A \( \, \cap \, \) B) would be drawing a red queen (either the queen of hearts or the queen of diamonds).

The probability of the intersection of two events is especially useful when calculating the union of events, as it ensures we don't count the common outcomes twice:
\[ P(A \, \cup \, B) = P(A) + P(B) - P(A \, \cap \, B) \]

In our exercise, we are given \( P(A \, \cap \, B) = 0.15 \), which represents the overlap where both A and B occur. This value is subtracted from the sum of the individual probabilities to provide the correct union probability.

Probability Formula

The probability formula is a vital tool in probability theory and statistics. It allows us to quantify how likely an event is to occur. Essentially, it provides a way to measure uncertainty and make informed predictions based on available data.

When dealing with multiple events, formulas such as the one for the union of events become crucial. The union probability formula is given by:
\[ P(A \, \cup \, B) = P(A) + P(B) - P(A \, \cap \, B) \]

This formula tells us how to calculate the probability that either event A happens, event B happens, or both occur. Each component serves a specific purpose:

• \( P(A) \) is the probability of event A happening.
• \( P(B) \) is the probability of event B happening.
• \( P(A \, \cap \, B) \) is the probability of both events happening.

By combining these elements, we can find the total probability, accounting for any overlap between the events. This is crucial for avoiding double-counting and ensuring accurate probabilities.

In our provided example, substituting the known values into the formula allowed us to find that \( P(A \, \cup \, B) = 0.55 \). This serves as a practical demonstration of how the probability formula is applied to solve real-world problems efficiently.