Question

If \(A\) and \(B\) are mutually exclunive events then \(P(A \cup B)=\ldots\)

Short answer

\(P(A \cup B) = P(A) + P(B)\)

Step-by-step solution

Define Mutually Exclusive Events

Two events, \(A\) and \(B\), are described as mutually exclusive. This means that both events cannot happen at the same time; in other words, \(A \cap B = \emptyset\). Therefore, the probability of both events occurring together is zero, i.e., \(P(A \cap B) = 0\).

Use the Probability Addition Rule

The general rule for the probability of the union of any two events \(A\) and \(B\) is given by the formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). For mutually exclusive events, \(P(A \cap B) = 0\). Substitute this into the formula.

Simplify the Formula for Mutually Exclusive Events

Substitute \(P(A \cap B) = 0\) into the expression: \[P(A \cup B) = P(A) + P(B) - 0\]. This simplifies to \(P(A \cup B) = P(A) + P(B)\).

Finalize the Solution

The probability of the union of two mutually exclusive events \(A\) and \(B\) is simply the sum of their individual probabilities. Therefore, \(P(A \cup B)\) is \(P(A) + P(B)\).

Key concepts

Mutually Exclusive Events

In probability, when we say two events are mutually exclusive, we mean that the two events cannot occur simultaneously. Imagine you are rolling a six-sided die. If one event is rolling a 3 and the other event is rolling a 4, these two events are mutually exclusive. You can't roll a 3 and a 4 in a single throw of the die at the same time.

To express this idea mathematically, if events \(A\) and \(B\) are mutually exclusive, then \(A \cap B = \emptyset\). This means the intersection of \(A\) and \(B\) is an empty set, or in terms of probability, \(P(A \cap B) = 0\). Recognizing mutually exclusive events is key to simplifying probability problems involving their union.

Probability Addition Rule

The probability addition rule helps us calculate the likelihood of either of two events occurring. The basic principle is that to find the probability of either event \(A\) or event \(B\) happening, we add together the probability of each event. If the events are not mutually exclusive, we must subtract the probability of both events occurring together, as it would be counted twice.

The formula is:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

For mutually exclusive events, because the probability of both happening together is zero, the formula simplifies to just the sum of their individual probabilities:

• \(P(A \cup B) = P(A) + P(B)\)

This is a simpler case where the subtraction of \(P(A \cap B)\) is unnecessary.

Union of Events

The union of two events, represented as \(A \cup B\), refers to the scenario where at least one of the events occurs. In practical terms, if you have two events, such as drawing a heart or a spade from a deck of cards, the union \(A \cup B\) is the probability of drawing either a heart or a spade.

For mutually exclusive events, calculating the union becomes straightforward. Since the events cannot occur together, the union is simply the sum of the probabilities of each event happening individually. This reduces the complexity when determining the likelihood of either event in cases of mutual exclusivity.

Remember, the union covers any situation where one or both events can happen, but with mutually exclusive events, it excludes any possibility of overlap, confirming \(P(A \cap B) = 0\). The union thus offers a comprehensive picture of combined probabilities under the constraint that no overlap occurs.