Problem 49 - Algebra and Trigonometry

Question

$$\begin{array}{l}\text { If } P(A)=0.60, P(A \cup B)=0.85, \text { and } P(A \cap B)=0.05 \\\\\text { find } P(B) .\end{array}$$

Step-by-Step Solution

Answer

0.30

Step 1: Understand the Given Information

First, identify and understand the values given in the problem: \(P(A) = 0.60\), \(P(A \cup B) = 0.85\), and \(P(A \cap B) = 0.05\). We need to find \(P(B)\).

Step 2: Apply the Formula for Union of Two Events

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

Step 3: Substitute the Given Values into the Formula

Substitute the given values into the formula: \(0.85 = 0.60 + P(B) - 0.05\).

Step 4: Solve for \(P(B)\)

Isolate \(P(B)\) by combining like terms and solving the equation: \(0.85 = 0.55 + P(B)\). Then, subtract 0.55 from both sides to find \(P(B)\): \(P(B) = 0.30\).

Key Concepts

Union of Events

A union of events in probability refers to the occurrence of at least one of the specified events. When we talk about the union of events A and B, denoted as \(A \cup B\), we mean that either event A happens, event B happens, or both happen.

To determine the probability of the union of two events, we use the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This formula helps us to account for the fact that if both events A and B occur together, we shouldn't count their overlap twice.

Intersection of Events

The intersection of events deals with the probability that both specified events will occur simultaneously. When discussing the intersection of events A and B, symbolized as \(A \cap B\), it means that both events A and B happen at the same time.

In the formula for the union of events, the term \(P(A \cap B)\) already appears, which represents this overlap or intersection. For example, in the given problem, \(P(A \cap B) = 0.05\), indicating that there's a 5% chance that both events A and B will happen together.

Probability Formulas

Probability formulas are mathematical tools used to quantify the likelihood of one or more events occurring. They often incorporate various elements such as addition, subtraction, and sometimes multiplication of individual probabilities.

The key formula used in the given exercise is for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Other important probability formulas include:

• Complementary Events: \(P(A') = 1 - P(A)\)
• Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
• Mutually Exclusive Events: \(P(A \cup B) = P(A) + P(B)\) if \(A \cap B = \emptyset\)

Set Theory in Probability

Set theory forms the foundation of probability theory. It helps in understanding and visualizing events and their relationships in probabilistic terms.

Basic concepts from set theory used in probability include:

• Universal Set: Represents all possible outcomes.
• Subset: A set contained within another set.
• Union: Combining two sets to include all distinct elements from both sets.
• Intersection: Common elements between sets.
• Empty Set: A set with no elements, useful in representing impossible events.

This theory allows clearer representations through Venn diagrams, aiding in solving complex probability problems by visualizing unions and intersections of events.