The
addition rule helped us solve problems when we performed one task and wanted to know the
probability of two things happening during that task. This lesson deals with the multiplication rule. The multiplication rule also deals with two events, but in these problems the events occur as a result of more than one task (rolling one die then another, drawing two cards, spinning a spinner twice, pulling two marbles out of a bag, etc).
When asked to find the
probability of A and B, we want to find out the
probability of events A and B happening.
The Multiplication Rule:
Consider events A and B. P(A
B)= P(A)
P(B).
Note: Some books will say to take care that A and B are independent, but the rule can also be used with dependent events, you just have to be more careful in find P(A) and P(B).
What The Rule Means:
Suppose we roll one die followed by another and want to find the
probability of rolling a 4 on the first die and rolling an even number on the second die. Notice in this problem we are not dealing with the sum of both dice. We are only dealing with the
probability of 4 on one die only and then, as a separate event, the
probability of an even number on one die only.
P(4) = 1/6
P(even) = 3/6
So P(4
even) = (1/6)(3/6) = 3/36 = 1/12
While the rule can be applied regardless of dependence or independence of events, we should note here that rolling a 4 on one die followed by rolling an even number on the second die are independent events. Each die is treated as a separate thing and what happens on the first die does not influence or effect what happens on the second die. This is our basic definition of independent events: the outcome of one event does not influence or effect the outcome of another event.
We’ll look at examples later that deal with dependent events. Just keep in mind that what happens on one event will effect the other event.
Let's Practice:
- Suppose you have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. You are going to pull out one marble, record its color, put it back in the box and draw another marble. What is the probability of pulling out a red marble followed by a blue marble?
The multiplication rule says we need to find P(red)
P(blue).
P(red) = 2/9
P(blue) = 3/9
P(red
blue) = (2/9)(3/9) = 6/81 = 2/27
The events in this example were independent. Once the first marble was pulled out and its color recorded, it was returned to the box. Therefore, the
probability for the second marble was not effected by what happened on the first marble.
Notice that the final answer is always simplified. Some students find it helpful to simplify before multiplying, but the final answer must always be simplified.
- Consider the same box of marbles as in the previous example. However in this case, we are going to pull out the first marble, leave it out, and then pull out another marble. What is the probability of pulling out a red marble followed by a blue marble?
We can still use the multiplication rule which says we need to find P(red)
P(blue). But be aware that in this case when we go to pull out the second marble, there will only be 8 marbles left in the bag.
P(red) = 2/9
P(blue) = 3/8
P(red
blue) = (2/9)(3/8) = 6/72 = 1/12
The events in this example were dependent. When the first marble was pulled out and kept out, it effected the
probability of the second event. This is what is meant by dependent events.
- Suppose you are going to draw two cards from a standard deck. What is the probability that the first card is an ace and the second card is a jack (just one of several ways to get “blackjack” or 21).
Using the multiplication rule we get
P(ace)
P(jack) = (4/52)(4/51) = 16/2652 = 4/663
Notice that this will be the same
probability even if the question had asked for the
probability of a jack followed by an ace.