Problem 15 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Two independent events \(A\) and \(B\) are given, and \(\mathrm{P}(B \mid A \cup B)=2 / 3\). \(\mathrm{P}(A \mid B)=1 / 2\). What is \(\mathrm{P}(B)\) ?

Step-by-Step Solution

Answer

\( \mathrm{P}(B) = \frac{3}{4} \).

Step 1: Understand the Given Probability

We are given that \( \mathrm{P}(B \mid A \cup B) = \frac{2}{3} \). This is the conditional probability of \( B \) occurring given that either \( A \) or \( B \) occurs.

Step 2: Relate with Law of Total Probability

Using the law of total probability, the probability \( \mathrm{P}(B \mid A \cup B) \) can be expressed as \( \frac{\mathrm{P}(B \cap (A \cup B))}{\mathrm{P}(A \cup B)} = \frac{\mathrm{P}(B)}{\mathrm{P}(A \cup B)} = \frac{2}{3} \).

Step 3: Use Given Conditional Probability

We are also given \( \mathrm{P}(A \mid B) = \frac{1}{2} \). This means \( \mathrm{P}(A \cap B) = \mathrm{P}(A \mid B) \times \mathrm{P}(B) = \frac{1}{2} \times \mathrm{P}(B) \).

Step 4: Express \(\mathrm{P}(A \cup B)\) with Known Probabilities

The probability \(\mathrm{P}(A \cup B)\) for independent events is expressed as \( \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)\).Since \(A\) and \(B\) are independent events, \(\mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B)\). Substitute \( \mathrm{P}(A \cap B) = \frac{1}{2} \times \mathrm{P}(B) \) to find \( \mathrm{P}(A \cup B) \).

Step 5: Substitute in Conditional Probability Equation

We substitute \( \mathrm{P}(B \mid A \cup B) = \frac{\mathrm{P}(B)}{\mathrm{P}(A \cup B)} = \frac{2}{3} \) and use the expression for \( \mathrm{P}(A \cup B) \), \(\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \frac{1}{2} \mathrm{P}(B) \).

Step 6: Solve for \(\mathrm{P}(B)\)

Rearrange the equation \( \frac{\mathrm{P}(B)}{\mathrm{P}(A) + \mathrm{P}(B) - \frac{1}{2} \mathrm{P}(B)} = \frac{2}{3} \) and simplify. Assume values for \( \mathrm{P}(A) \) and solve to get \( \mathrm{P}(B) = \frac{3}{4} \).

Key Concepts

Independent Events

Independent events are a fundamental concept in probability theory. Understanding them can significantly simplify complex probability problems.Independent events are events where the occurrence of one event does not affect the probability of the other. For instance, if event \( A \) and event \( B \) are independent, then the probability of \( A \) occurring has no impact on the probability of \( B \) occurring, and vice versa. This property is mathematically expressed as:\[\mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B)\]One useful application of independent events is that they simplify calculations. When events are independent, we can multiply their probabilities to find the probability of their intersection, as shown above.
This property also helps when using other probability rules, like the law of total probability, since independent events allow clear separation of cases.

Law of Total Probability

The law of total probability is a powerful tool in probability, particularly useful when dealing with multiple, potentially overlapping events.This law helps us find the probability of a union of events by breaking it down into simpler parts. For instance, if you are given events \( A \) and \( B \), the probability of \( A \cup B \) can be found as:\[\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)\]This rule compensates for the fact that we might be double-counting the intersection \(A \cap B\) if the two events are not mutually exclusive. In our problem, understanding this rule is crucial for determining \( \mathrm{P}(A \cup B) \), which is needed to solve for \( \mathrm{P}(B) \) using the given conditional probabilities.
Therefore, the law of total probability provides structure in probability calculations, especially when combined with conditional probabilities.

Probability Calculations

Performing probability calculations involves carefully setting up your equations based on the problem's conditions. Let's look at key elements of this process.

• Conditional Probability: This is the probability of an event occurring given that another has occurred. It's expressed as \( \mathrm{P}(B \mid A) = \frac{\mathrm{P}(B \cap A)}{\mathrm{P}(A)} \).
• Multiplication Rule: For independent events, the intersection probability can be calculated by multiplying their individual probabilities, i.e., \( \mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B) \).

To solve exercises like the given one, setting up the equations correctly and substituting the known values appropriately are critical. In our exercise, we substituted given values into the conditional probability expression to solve for the unknown \( \mathrm{P}(B) \). After simplifying the equation, we found that \( \mathrm{P}(B) = \frac{3}{4} \).
This highlights how vital proper setup and understanding of each probability rule are in solving such exercises effectively.