Question

Suppose that \(P(A)=.3\) and \(P(B)=.5 .\) If events \(A\) and \(B\) are mutually exclusive, find these probabilities: a. \(P(A \cap B)\) b. \(P(A \cup B)\)

Short answer

Answer: For mutually exclusive events A and B, a. The probability of their intersection, P(A ∩ B), is 0. b. The probability of their union, P(A ∪ B), is 0.8.

Step-by-step solution

Understand the properties of mutually exclusive events

When two events are mutually exclusive, it means that they cannot occur at the same time. In other words, if one of them occurs, the other cannot. Mathematically, it means that the intersection of the two events is the empty set, i.e., \(P(A \cap B) = 0\).

Find the probability of A intersection B

Since A and B are mutually exclusive, the probability of their intersection is 0. Therefore, \(P(A \cap B) = 0\).

Find the probability of A union B

To find the probability of the union of A and B, we can use the formula for the probability of the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Since we already know that \(P(A \cap B) = 0\), we can simplify this formula as \(P(A \cup B) = P(A) + P(B)\). Plug the given values into the formula: \(P(A \cup B) = P(A) + P(B) = 0.3 + 0.5 = 0.8\). The probabilities for these events are: a. \(P(A \cap B) = 0\) b. \(P(A \cup B) = 0.8\)

Key concepts

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur simultaneously. Think of these events like two separate doors; choosing one automatically excludes the other. In probability terms, if two events are mutually exclusive, the probability of them both occurring at the same time is zero. This is mathematically represented as \( P(A \cap B) = 0 \).

An example to visualize this is flipping a coin—obtaining a "heads" and a "tails" in one flip is impossible, hence mutually exclusive. Understanding this principle is important because it helps us in calculating probabilities related to these events correctly.

Probability of Intersection

The probability of intersection between two events refers to the likelihood that both events will happen at the same time. In mathematical terms, this is defined by \( P(A \cap B) \).

For mutually exclusive events, as mentioned earlier, this probability is zero because they cannot occur together. However, if events were not mutually exclusive, an example might include drawing a red card and a king from a deck of cards—the king can also be red. In scenarios where events aren't mutually exclusive, intersections are calculated considering common elements between events.

Knowing how to calculate intersections allows us to dive deeper into more complex probability scenarios.

Probability of Union

The probability of the union of two events refers to the probability that at least one of the events will occur. It’s symbolized by \( P(A \cup B) \). The formula is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

In simpler terms, this formula helps to calculate the chances of either event A happening, event B happening, or both, ensuring not to double-count the overlap if they intersect. For mutually exclusive events, since their intersection is zero, the formula simplifies to \( P(A \cup B) = P(A) + P(B) \).

This union concept extends to broader applications, such as calculating the probability of at least one success in multiple independent trials, forming the foundation for more advanced statistical methods.