Addition Rule of Probability - Explained

The addition rule of probability is used to answer questions about the likelihood of at least one of multiple criteria being true, or the probability of event A or event B occurring. This rule is applicable to various scenarios, such as rolling a 2 or a 5, drawing an 8 or a spade, or owning an Android or an iPhone.

For mutually exclusive events, which cannot occur simultaneously (e.g., rolling a 2 or a 5 in a single roll), the probability of at least one occurring is the sum of their individual probabilities. For instance, the probability of rolling a 2 (1/6) or a 5 (1/6) is 2/6, by simply adding the probabilities together.

When events are not mutually exclusive (e.g., drawing a heart or a face card, as a queen of hearts is both), the formula is adjusted. After adding the individual probabilities, you must subtract the probability of both events occurring simultaneously to avoid double-counting. This can be visualized with a Venn diagram where the overlap is subtracted.

Consider a group of students where 50% play soccer, 20% play tennis, and 10% play both. To find the probability that a student plays at least one sport (soccer or tennis), you add the probability of playing soccer (0.50) to the probability of playing tennis (0.20) and subtract the probability of playing both (0.10). The result is a 60% chance that a student plays at least one of the two sports.

In summary, the addition rule of probability is a fundamental tool for determining the likelihood of event A or event B happening, or similarly, the probability that at least one of several events will occur. The key is to distinguish between mutually exclusive and non-mutually exclusive events to apply the correct formula.