Final answer:
The multiplication theory of probability, or 'And/Then' rule, indicates that the probability of two independent events both occurring is the product of their individual probabilities, expressed as P(A AND B) = P(A) * P(B). For dependent events, the formula is P(A AND B) = P(A) * P(B|A). This concept is distinct from the addition rule, which applies to mutually exclusive events.
Step-by-step explanation:
The multiplication theory of probability, also known as the 'And/Then' rule, is a fundamental concept used to determine the likelihood of two events occurring together. Specifically, when considering two independent events, the rule states that the probability of both events occurring is equal to the product of their individual probabilities. This is formally expressed as P(A AND B) = P(A) * P(B), provided that events A and B are independent.
In contrast, for dependent events where the outcome of one can affect the other, the rule is adjusted to account for this dependency: P(A AND B) = P(A) * P(B|A), wherein P(B|A) represents the probability of event B occurring given that event A has already happened. It is important to recognize that this formula applies when events A and B are not mutually exclusive. On the other hand, the addition rule is employed when dealing with mutually exclusive events, where the chance of either event occurring is the sum of their individual probabilities: P(A OR B) = P(A) + P(B) — albeit with adjustments made when events are not mutually exclusive.
The multiplication theory of probability is widely applicable, from predicting the possible outcomes of genetic crosses in biology to calculating the odds of complex events in finance. Understanding when to use the multiplication rule versus the addition rule is crucial for solving a variety of problems in probability and statistics.