What is the general multiplication rule?

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The general multiplication rule states that the probability of any two events, A and B, both happening can be calculated as:

P(A and B) = P(A) * P(B|A)

The vertical bar | means “given.” Thus, P(B|A) can be read as “the probability that B occurs, given that A has occurred.”

If events A and B are independent, then P(B|A) is simply equal to P(B) and the rule can be simplified to:

P(A and B) = P(A) * P(B)

Let’s walk through a few examples of both independent and dependent events to see how we can apply this general multiplication rule in practice.

The General Multiplication Rule for Dependent Events

The following examples illustrate how to use the general multiplication rule to find probabilities related to two dependent events. In each example, the probability that the second event occurs is affected by the outcome of the first event.

Example 1: Balls in an Urn

An urn contains 4 red balls and 3 green balls. Bob is going to randomly select 2 balls from the urn, without replacement. What is the probability that he chooses 2 red balls?

Solution: The probability that he selects a red ball on the first attempt is 4/7. Once that ball is then removed, the probability that he selects a red ball on the second attempt is 3/6. Thus, the probability that he selects 2 red balls can be calculated as:

P(Both red) = 4/7 * 3/7 ≈ 0.2249

Example 2: Cards in a Deck

A deck of cards contains 26 black cards and 26 red cards. Debbie is going to randomly select 2 cards from the deck, without replacement. What is the probability that she chooses 2 red cards?

Solution: The probability that she selects a red card on the first attempt is 26/52. Once that card is then removed, the probability that she selects a red card on the second attempt is 25/51. Thus, the probability that she selects 2 red cards can be calculated as:

P(Both red) = 26/52 * 25/51 ≈ 0.2451

The General Multiplication Rule for Independent Events

Example 1: Flipping Two Coins

Suppose we flip two coins. What is the probability that both coins land on heads?

Solution: The probability that the first coin lands on heads is 1/2. No matter which side the first coin lands on, the probability that the second coin lands on heads is also 1/2. Thus, the probability that both coins land on heads can be calculated as:

P(Both land on heads) = 1/2 * 1/2 = 0.25

Example 2: Rolling Two Dice

Suppose we roll two dice at once. What is the probability that both dice land on the number 1?

Solution: The probability that the first dice lands on “1” is 1/6. No matter which side the first dice lands on, the probability that the second dice lands on “1” is also 1/6. Thus, the probability that both dice land on “1” can be calculated as:

P(Both land on “1”) = 1/6 * 1/6 = 1/36 ≈ 0.0278