Multiplication Rule for Independent Events

It is important to know how to calculate the probability of an event. Certain types of events in probability are called independent. When we have a pair of independent events, sometimes we may ask, "What is the probability that both of these events occur?" In this situation, we can simply multiply our two probabilities together.

We will see how to utilize the multiplication rule for independent events. After we have gone over the basics, we will see the details of a couple of calculations.

Definition of Independent Events

We begin with a definition of independent events. In probability, two events are independent if the outcome of one event does not influence the outcome of the second event.

A good example of a pair of independent events is when we roll a die and then flip a coin. The number showing on the die has no effect on the coin that was tossed. Therefore these two events are independent.

An example of a pair of events that are not independent would be the gender of each baby in a set of twins. If the twins are identical, then both of them will be male, or both of them would be female.

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Statement of the Multiplication Rule

The multiplication rule for independent events relates the probabilities of two events to the probability that they both occur. In order to use the rule, we need to have the probabilities of each of the independent events. Given these events, the multiplication rule states the probability that both events occur is found by multiplying the probabilities of each event.

Formula for the Multiplication Rule

The multiplication rule is much easier to state and to work with when we use mathematical notation.

Denote events A and B and the probabilities of each by P(A) and P(B). If A and B are independent events, then:

P(A and B) = P(A) x P(B)

Some versions of this formula use even more symbols. Instead of the word "and" we can instead use the intersection symbol: ∩. Sometimes this formula is used as the definition of independent events. Events are independent if and only if P(A and B) = P(A) x P(B).

Example #1 of the Use of the Multiplication Rule

We will see how to use the multiplication rule by looking at a few examples. First suppose that we roll a six sided die and then flip a coin. These two events are independent. The probability of rolling a 1 is 1/6. The probability of a head is 1/2. The probability of rolling a 1 and getting a head is 1/6 x 1/2 = 1/12.

If we were inclined to be skeptical about this result, this example is small enough that all of the outcomes could be listed: {(1, H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T), (5, T), (6, T)}. We see that there are twelve outcomes, all of which are equally likely to occur. Therefore the probability of 1 and a head is 1/12. The multiplication rule was much more efficient because it did not require us to list our the entire sample space.

Example #2 of the Use of the Multiplication Rule

For the second example, suppose that we draw a card from a standard deck, replace this card, shuffle the deck and then draw again. We then ask what is the probability that both cards are kings. Since we have drawn with replacement, these events are independent and the multiplication rule applies. 

The probability of drawing a king for the first card is 1/13. The probability for drawing a king on the second draw is 1/13. The reason for this is that we are replacing the king that we drew from the first time. Since these events are independent, we use the multiplication rule to see that the probability of drawing two kings is given by the following product 1/13 x 1/13 = 1/169.

If we did not replace the king, then we would have a different situation in which the events would not be independent. The probability of drawing a king on the second card would be influenced by the result of the first card.