probability of mutually exclusive events
Check the final answer first, then review the worked steps.
Problem
probability of mutually exclusive events
Answer
\(9/20\)
Step-by-step solution
- Understand the definition of mutually exclusive events: Two events $A$ and $B$ are mutually exclusive if they cannot occur at the same time. This means that the probability of both events occurring simultaneously is zero, i.e., $P(A \cap B) = 0$.
- Apply the Addition Rule for probability: The probability of either event $A$ or event $B$ occurring is given by the formula: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
- Substitute the given values: Since the events are mutually exclusive, $P(A \cap B) = 0$. We are given $P(A) = \frac{1}{4}$ and $P(B) = \frac{1}{5}$. Substituting these into the formula, we get: $$P(A \cup B) = \frac{1}{4} + \frac{1}{5} - 0$$
- Calculate the sum: To add the fractions, find a common denominator, which is $20$: $$P(A \cup B) = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}$$
- Convert to decimal (optional): The fraction $\frac{9}{20}$ can also be expressed as $0.45$.