Problem 48 - Algebra and Trigonometry

Question

Find the probability of the indicated event if \(P(A)=0.25\) and \(P(B)=0.45\) $$P(A \cap B) \text { if } A, B \text { are mutually exclusive }$$

Step-by-Step Solution

Answer

The probability is 0.

Step 1: Understand Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. This means if event A occurs, event B cannot occur, and vice versa.

Step 2: Define the Intersection of A and B

For mutually exclusive events, the probability of both events occurring at the same time, denoted as \(P(A \cap B)\), is 0. This is because the occurrence of one event excludes the occurrence of the other.

Step 3: Conclusion

Based on the definition of mutually exclusive events, we can conclude that \(P(A \cap B) = 0\).

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur simultaneously. This means if one event happens, the other cannot. Imagine rolling a die; getting a 1 and getting a 2 at the same roll are mutually exclusive because both outcomes can't happen at once. Understanding this helps in calculating correct probabilities by acknowledging restrictions. These events have no overlap or common outcomes.

In mathematical terms, for two events A and B, if they are mutually exclusive, then the probability of both happening together is zero. That is, \(\text{P}(A \cap B) = 0\).

Intersection of Events

In probability, the intersection of two events A and B, denoted as \(\text{P}(A \cap B)\), represents the probability that both events happen at the same time. For non-mutually exclusive events, this can be found using various probability rules.

However, when events are mutually exclusive, the intersection is zero because the events cannot coincide. The mathematical definition here simplifies our calculations: \(\text{P}(A \cap B) = 0\) when A and B are mutually exclusive. This means their paths do not cross in a probability scenario.

In simpler words, mutually exclusive events have no common outcomes, making their intersection null.

Probability Rules

Understanding core probability rules clears up how to work with different event scenarios.

1. **Addition Rule for Mutually Exclusive Events:** The probability of either event A or event B occurring is the sum of their individual probabilities, since they cannot occur together. Mathematically, \(\text{P}(A \cup B) = \text{P}(A) + \text{P}(B)\).

2. **Addition Rule for Non-Mutually Exclusive Events:** Here, since events can overlap, we must subtract the intersection to avoid double-counting: \(\text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B)\).

3. **Multiplication Rule:** For independent events, the probability that both events A and B occur is the product of their individual probabilities, expressed as \(\text{P}(A \cap B) = \text{P}(A) \times \text{P}(B)\).

The exercise involves mutually exclusive events, thus focusing on specific rules without intersection considerations provides clarity on probability calculations.