Question

If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A|B) = \frac{2}{5} \), then find \( P(A \cup B) \).

Hint:

When solving probability problems, always list the given information and identify the target variable. Then, write down the relevant formulas that connect the knowns to the unknown. This systematic approach prevents confusion and errors.

Answer 1

Step 1: Understanding the Concept:
This question requires the use of fundamental probability rules, specifically the formula for conditional probability and the addition rule for probabilities.
Step 2: Key Formula or Approach:
1. Conditional Probability: \( P(A/B) = \frac{P(A \cap B)}{P(B)} \)
2. Addition Rule: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
Step 3: Detailed Explanation:
Part 1: Find \( P(A) \) and \( P(B) \).
We are given \( 2P(A) = P(B) = \frac{5}{13} \).
From this, we can deduce:
\[ P(B) = \frac{5}{13} \] \[ 2P(A) = \frac{5}{13} \implies P(A) = \frac{1}{2} \times \frac{5}{13} = \frac{5}{26} \] Part 2: Find \( P(A \cap B) \).
Using the conditional probability formula: \( P(A/B) = \frac{P(A \cap B)}{P(B)} \).
We are given \( P(A/B) = \frac{2}{5} \) and we know \( P(B) = \frac{5}{13} \).
Rearranging the formula to solve for \( P(A \cap B) \):
\[ P(A \cap B) = P(A/B) \times P(B) \] \[ P(A \cap B) = \frac{2}{5} \times \frac{5}{13} = \frac{10}{65} = \frac{2}{13} \] Part 3: Find \( P(A \cup B) \).
Now use the addition rule for probabilities:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the values we found:
\[ P(A \cup B) = \frac{5}{26} + \frac{5}{13} - \frac{2}{13} \] To add/subtract these fractions, find a common denominator, which is 26.
\[ P(A \cup B) = \frac{5}{26} + \frac{5 \times 2}{13 \times 2} - \frac{2 \times 2}{13 \times 2} \] \[ P(A \cup B) = \frac{5}{26} + \frac{10}{26} - \frac{4}{26} \] \[ P(A \cup B) = \frac{5 + 10 - 4}{26} = \frac{11}{26} \] Step 4: Final Answer:
The value of \( P(A \cup B) \) is \( \frac{11}{26} \).