Understanding Independent Events
When we say two events A and B are independent, it means that the occurrence of one event does not affect the occurrence of the other.
Given Probabilities
- P(A) = 3/5
- P(B) = 2/3
Calculating P(A and B)
For independent events, the probability of both events A and B occurring, denoted as P(A and B), is calculated using the formula:
P(A and B) = P(A) * P(B)
Substituting the Values
- P(A and B) = (3/5) * (2/3)
Carrying Out the Multiplication
- Multiply the numerators: 3 * 2 = 6
- Multiply the denominators: 5 * 3 = 15
Thus,
P(A and B) = 6/15
This fraction can be simplified:
P(A and B) = 2/5
Conclusion
In summary, when events A and B are independent, the probability of both occurring together is 2/5. This shows how the independence of events allows us to multiply their probabilities directly, leading to a straightforward calculation. Understanding this concept is crucial in probability theory, especially in fields like statistics, finance, and risk assessment.