The Probability of Compound Events - The Multiplication Rule

Compound events occur when two or more simple events occur.

If we are to assess the probability of whether event A or event B will occur, we need to use the Addition Rule discussed in an earlier article. In this article we will discuss the probability of event A and Event B occurring.
This implies that these events occur in sequence. Let's look at the example of tossing a dime and a nickel.
What is the probability of getting heads on the nickel and a heads on the dime? The probability of getting a heads on the nickel, which is 1, divided by 2 or one half.
The probability of getting a heads on the dime is also I divided by 2, for a value of one half.
To determine the probability of getting a head on the nickel and the dime, we multiply the two probabilities for a result of 1 over 4, or one fourth.

Let's also note that these two events are Independent.

This means that the outcome of one event has no dependence or influence on the outcome of the other event.
That is, the result of tossing the dime has no impact on the result of tossing the nickel.

Let's look at another case, here we have ten marbles, five blue and five red.

They are place in an urn and one is chosen at random.
What is the probability of choosing a red one? It is the number of ways we could choose a red marble, which are five, divided by the number of possible outcomes, which are ten. This makes the probability of randomly choosing a red one equal to one half.
Now given that we have already chosen a red marble, what is the probability of choosing a second red one? It is the number of ways we could choose a red one, which is now four, because there are only four red marbles remaining in the urn, divided by the number of possible outcomes, which is now nine, because there are only nine marbles remaining in the urn. So the probability of choosing a second red marble is four divided by nine.
This makes the probability of choosing a red marble, followed by a second red marble equal to one half times four ninths, which is 4 over 18, or 2 over 9. These two events are Dependent, that is, the probability of the second event depends on the outcome of the first event.
Remember the Multiplication Rule is used to calculate the probability of the occurrence of two events.