The AND/OR Rules

The AND/OR Rules

The AND Rule in Probability

  • The AND rule in probability relates to when we are interested in the probability of two events happening at the same time, often represented with the notation P(A ∩ B). This is interpreted as the probability of event A and event B both occurring.

  • For independent events, the probability of both events happening is calculated by multiplying the probabilities of the individual events. This can be represented as P(A ∩ B) = P(A) * P(B) when A and B are independent.

  • The AND rule can also be used for dependent events. In this case, we multiply the probability of the first event by the conditional probability of the second event given that the first has occurred. This is represented as **P(A ∩ B) = P(A) * P(B A)**.

The OR Rule in Probability

  • The OR rule relates to when we are interested in the probability of either one event or another occurring, represented using the notation P(A ∪ B). This is interpreted as the probability of event A or event B (or both) occurring.

  • To calculate this, we add together the probabilities of the individual events. If the events are not mutually exclusive, meaning they can occur at the same time, we then subtract the probability of both events occurring. This can be represented as P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

  • In the case where events A and B are mutually exclusive, meaning they can’t occur at the same time, then the probability of either event occurring is simply the sum of their individual probabilities. Therefore, we have P(A ∪ B) = P(A) + P(B) for mutually exclusive events.

Intersection and Union

  • When using the AND and OR rules, it’s important to understand the concepts of intersection and union. Intersection (A ∩ B) refers to events that both A and B share, while Union (A ∪ B) refers to all outcomes that belong to A or B or both.

  • For example, if we roll a die once, the event A could be rolling a number less than 4 (1, 2, 3) and event B could be rolling an odd number (1, 3, 5). The intersection of A and B consists of outcomes they both share (1, 3), while the union of A and B includes all outcomes that belong to A or B or both (1, 2, 3, 5).

  • Being able to distinguish between union and intersection is critical when applying the OR and AND rules in solving probability problems.