Problem 54 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are mutually exclusive and \(P(B) \neq 0\), then \(P(A \mid B)=0\)

Step-by-Step Solution

Answer

The statement is true. Since events A and B are mutually exclusive, they cannot both occur at the same time, meaning P(A ∩ B) = 0. Using the conditional probability formula, \(P(A \mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0}{P(B)}\). As P(B) ≠ 0, \(P(A \mid B) = \frac{0}{P(B)} = 0\).

Step 1: Conditional Probability Formula

Recall the formula for conditional probability: \(P(A \mid B)=\frac{P(A\cap B)}{P(B)}\).

Step 2: Mutually Exclusive Events

Events A and B are mutually exclusive, which means they cannot both occur at the same time. In other words, their intersection is empty, so P(A ∩ B) = 0.

Step 3: Substitute into the Formula

Substitute P(A ∩ B) = 0 into the conditional probability formula: \(P(A \mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0}{P(B)}\).

Step 4: Final Result

Since P(B) ≠ 0 and 0 divided by any nonzero number is 0, we have \(P(A \mid B) = \frac{0}{P(B)} = 0\). Thus, the statement is true: If A and B are mutually exclusive and P(B) ≠ 0, then P(A | B) = 0.

Key Concepts

Mutually Exclusive Events

In probability theory, mutually exclusive events are events that cannot happen at the same time. Imagine flipping a coin – it can land either on heads or tails, but not both. This is a classic example of mutually exclusive events. If you have two events, say A and B, and they are mutually exclusive, then the occurrence of one prevents the other from happening.
Because they don’t overlap, their intersection is empty. Mathematically, if A and B are mutually exclusive, we write this as follows:

• The intersection: \( P(A \cap B) = 0 \)

This implies that the probability of both A and B occurring simultaneously is zero. Understanding mutually exclusive events is crucial in probability theory as it aids in calculating overall probabilities accurately.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It’s like figuring out the chances of a rainy day given that you've already seen clouds forming in the sky. For two events A and B, the conditional probability of A given B is represented by \( P(A \mid B) \).
We use the formula:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)

This equation implies a requirement that \( P(B) eq 0 \) because we are using it as the denominator. If event A is impossible when event B has occurred, like in mutually exclusive events, the conditional probability will logically equal zero. The formula essentially tells us how probable event A is, under the condition that B has happened.

Intersection of Events

The intersection of events in probability refers to the scenario where two or more events occur simultaneously. For events A and B, their intersection, written as \( A \cap B \), represents outcomes that are common to both events.
In practical terms, imagine you’re in a situation where you want both a sunny day and no homework to do. The intersection here is that specific set of conditions where both happen at the same time. In mathematical expressions, if the events are mutually exclusive (as discussed earlier), then their intersection is the empty set:

• \( P(A \cap B) = 0 \)

This is because there are no outcomes common to both if they are mutually exclusive. Understanding intersections helps in calculating compound probabilities, understanding joint occurrences, and preventing errors in probability calculations.