Introduction to Independent Events
In probability theory, two events $A$ and $B$ are defined as independent if the occurrence of one event does not affect the probability of the other occurring. Mathematically, this is expressed as: $P(A \\cap B) = P(A) \\times P(B)$
The Addition Rule
To find the probability of the union of two events, $P(A \\cup B)$, we use the General Addition Rule: $P(A \\cup B) = P(A) + P(B) - P(A \\cap B)$
Step-by-Step Solution
Given the values from your problem:
- $P(A) = \\frac{2}{3}$
- $P(B) = \\frac{3}{5}$
Step 1: Calculate the intersection $P(A \\cap B)$
Since the events are independent: $P(A \\cap B) = P(A) \\times P(B) = \\frac{2}{3} \\times \\frac{3}{5} = \\frac{6}{15} = \\frac{2}{5}$
Step 2: Apply the Addition Rule
Substitute the values into the formula: $P(A \\cup B) = P(A) + P(B) - P(A \\cap B)$ $P(A \\cup B) = \\frac{2}{3} + \\frac{3}{5} - \\frac{2}{5}$
Find a common denominator (which is 15): $P(A \\cup B) = \\frac{10}{15} + \\frac{9}{15} - \\frac{6}{15}$ $P(A \\cup B) = \\frac{10+9-6}{15} = \\frac{13}{15}$
Thus, the probability $P(A \\cup B)$ is 13/15.