Problem 50 - Algebra and Trigonometry

Question

$$\begin{array}{l}\text { If } P(B)=0.30, P(A \cup B)=0.65, \text { and } P(A \cap B)=0.15 \\\\\text { find } P(A) .\end{array}$$

Step-by-Step Solution

Answer

P(A) = 0.50

Step 1: Understand the Given Information

The problem provides the following probabilities: - Probability of event B, written as: \( P(B) = 0.30 \)- Probability of either event A or event B (union), written as: \( P(A \,\cup\, B) = 0.65 \)- Probability of both events A and B occurring together (intersection), written as: \( P(A \,\cap\, B) = 0.15 \)

Step 2: Use the Formula for the Union of Two Events

Recall the formula for the union of two events, which is given by:\[ P(A \,\cup\, B) = P(A) + P(B) - P(A \,\cap\, B) \]This formula can be rearranged to solve for \( P(A) \).

Step 3: Insert the Given Values into the Formula

Substitute the given probabilities into the formula:\[ 0.65 = P(A) + 0.30 - 0.15 \]

Step 4: Solve for P(A)

Isolate \( P(A) \) by solving the equation:\[ 0.65 = P(A) + 0.15 \]Subtract 0.15 from both sides:\[ 0.65 - 0.15 = P(A) \]This simplifies to:\[ P(A) = 0.50 \]

Key Concepts

Union of Events

In probability, the union of two events, denoted as \(P(A \,\cup\, B)\), represents the probability that either event A occurs, event B occurs, or both events occur. The key formula for the union of two events is:
\[P(A \,\cup\, B) = P(A) + P(B) - P(A \,\cap\, B)\]
This formula helps us ensure we do not double-count the probability of both events occurring.
Example: If you know \(P(B) = 0.30\), \(P(A \,\cup\, B) = 0.65\), and \(P(A \,\cap\, B) = 0.15\), you can use the union of events formula to find \(P(A)\). First, rearrange the formula to solve for \(P(A)\). Substitute the given probabilities:
\[0.65 = P(A) + 0.30 - 0.15\]
Then, isolate \(P(A)\):
\[P(A) = 0.65 - 0.15 = 0.50\]
This means the probability of event A occurring is 0.50.

Intersection of Events

The intersection of two events, symbolized as \(P(A \,\cap\, B)\), refers to the probability that both events A and B happen at the same time. To understand this better, think of it as the overlap in a Venn diagram where both circles intersect.
Example: Suppose \(P(A \,\cap\, B) = 0.15\). This means there's a 15% chance that both events A and B occur together.
In our given problem, knowing the intersection value was crucial to finding \(P(A)\). We used the formula:
\[P(A \,\cup\, B) = P(A) + P(B) - P(A \,\cap\, B)\]
Substituting the known values allowed us to eliminate the double-counted area, thus accurately determining \(P(A)\).

Solving Equations

Solving equations is a fundamental skill in probability and many other areas of mathematics. It involves finding the value of a variable that makes an equation true. Let's break down how we solve the equation in this problem step-by-step:

• Start with the equation given by the formula \(P(A \,\cup\, B) = P(A) + P(B) - P(A \,\cap\, B)\).
• Insert the known values: \[0.65 = P(A) + 0.30 - 0.15.\]
• Combine like terms on the right-hand side: \[0.65 = P(A) + 0.15.\]
• Isolate \(P(A)\) by subtracting 0.15 from both sides: \[0.50 = P(A).\]

This process shows how inserting known values into a formula and then performing basic algebraic steps (such as addition or subtraction) can solve for an unknown probability. Understanding these foundational steps will boost your confidence in tackling more complex problems in the future.