Problem 59 - College Algebra

Question

Explain how to find or probabilities with mutually exclusive events. Give an example.

Step-by-Step Solution

Answer

The probability of 'or' with mutually exclusive events is simply the sum of the probabilities of each event. If Event A and Event B are mutually exclusive, then the probability that either A or B will occur is found using the formula: \(P(A \cup B) = P(A) + P(B)\). For instance, with a six-sided dice, where A is getting a 2, and B is getting a 5, the probability that either event A or event B would occur is \(1/3\).

Step 1: Definition of Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. In other words, if one event happens, the other one cannot. For example, throwing a dice and getting either 2 or 5.

Step 2: Definition of 'Or' in Probabilities

The term 'or' in probability refers to the occurrence of at least one of the events. For instance, when we say Event A or Event B, it means either A or B or both can occur.

Step 3: Finding 'Or' Probabilities with Mutually Exclusive Events

For mutually exclusive events A and B, the probability that either event A or event B will occur is given by the formula: \(P(A \cup B) = P(A) + P(B)\). The symbol '\(\cup\)' denotes the union of two sets, which corresponds to the 'or' in probabilities.

Step 4: Example of Mutually Exclusive Events and 'Or' Probabilities

Let's consider an example with a six-sided dice. Let's Event A be getting a 2, and Event B be getting a 5. These events are mutually exclusive as you can't get both on a single toss of the dice. Suppose you roll the dice one time, the probability of each event is \(1/6\). So, applying the formula, we get \(P(A \cup B) = P(A) + P(B) = 1/6 + 1/6 = 2/6 = 1/3\). Thus, the chance that either a 2 or a 5 will be rolled on a single toss of a die is \(1/3\).