Understanding the probability of multiple events is essential in probability theory, especially when distinguishing between events that can occur simultaneously and those that cannot. Events that cannot happen at the same time are termed mutually exclusive. For instance, if you consider the events of wearing a blue shirt and wearing a green shirt, these events are represented by separate circles in a Venn diagram, indicating that they cannot overlap; you can wear either one or the other, but not both at the same time.

In contrast, events that can occur together are not mutually exclusive. For example, wearing a blue shirt and green pants can happen simultaneously, as illustrated by overlapping circles in a Venn diagram. This distinction is crucial for calculating probabilities.

To calculate the probability of mutually exclusive events, you simply add their individual probabilities. This is often referred to as or probability, denoted by the symbol ∪ in set notation. For example, if you want to find the probability of rolling a 3 or a 5 on a six-sided die, you would calculate it as follows:

Let \( P(A) \) be the probability of rolling a 3, and \( P(B) \) be the probability of rolling a 5. Since there is one way to roll a 3 and one way to roll a 5, the probabilities are:

\( P(A) = \frac{1}{6} \) and \( P(B) = \frac{1}{6} \)

Adding these probabilities gives:

\( P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \)

This means the probability of rolling a 3 or a 5 is \( \frac{1}{3} \), which is approximately 0.33. Understanding these concepts allows for a clearer approach to calculating probabilities in various scenarios, enhancing your analytical skills in probability theory.