Multiplication Rules in Probability

Independent events

Events where the occurrence of one event does not affect the probability of the other event.

Dependent events

Events where the occurrence of one event affects the probability of the other event.

Multiplication rule for independent events

For two independent events E and F, the probability of both occurring is P(E) * P(F) with replacement.

P(E and F) for independent events

The probability of events E and F occurring is equal to P(E) × P(F).

Example of independent events probability calculation

The probability of drawing a king and a queen from a full deck is (4/52) * (4/52).

Multiplication rule for dependent events without replacement

For dependent events E and F, the probability of both occurring is P(E) * P(F|E).

P(E and F) for dependent events

The probability of events E and F occurring is P(E) * P(F given E).

Example of dependent events probability calculation

The probability of drawing a king and queen without replacement is (4/52) * (4/51).

Conditional probability for dependent events

The probability of event F occurring given event E occurs first is written as P(F|E).

Fundamental counting principle

For a multi-stage experiment, the total possible outcomes is the product of the outcomes at each stage: k1 * k2 * ... * kn.

Example of fundamental counting

To find the total number of students in a school with 5 grades, 5 classes per grade, and 20 students per class: 5 * 5 * 20 = 500.

Probability of queen given king

The probability of drawing a queen after already drawing a king is calculated as P(Q|K) = P(K) * P(Q|K) / P(K).