The Multiplication Rule of Probability
Probabilities of an Event and Another Event
Probability tells us how likely (how probable) it is an event will happen.
For example, it tells us that when a coin is tossed, the probability of the coin landing Heads up is ^{1}⁄_{2}.
It tells us that when a die is rolled, the probability of rolling a 6 is ^{1}⁄_{6}.
Imagine we wanted to find the probability of tossing Heads and rolling a 6.
The Multiplication Rule
To find a probability of one event and another event...
Probability of Heads and Probability of 6
...replace the and with a ×...
Probability of Heads × Probability of 6
This is the multiplication rule.
How to Find the Probability of an Event and Another Event
Question
What is the probability of getting Heads in a coin toss and rolling a 6 on a die?
StepbyStep:
1
Write down what we are trying to find out.
Probability of throwing a Heads and Probability of rolling a 6
2
Replace and with ×.
Probability of throwing a Heads × Probability of rolling a 6
3
Find the probability of tossing Heads.
The probability of tossing Heads is ^{1}⁄_{2}.
4
Find the probability of rolling a 6.
The probability of rolling a 6 is ^{1}⁄_{6}.
5
Substitute the probability of tossing Heads (^{1}⁄_{2}) and rolling a 6 (^{1}⁄_{6}) into the formula.
^{1}⁄_{2} × ^{1}⁄_{6} = ^{1}⁄_{12}
Answer:
The probability of tossing Heads and rolling a 6 is ^{1}⁄_{12}.
A Formula for the Multiplication Rule of Probability
The formula for finding the probability of event A and event B is shown below:
Let's use the formula in an example.
Question
What is the probability of rolling an odd number on a die and picking a Spade from a deck of cards?
StepbyStep:
1
Start with the formula.
P(A and B) = P(A) × P(B)
2
Define the events in our example.

Let O be the event of rolling an odd number. P(O) is the probability of rolling an odd number on the die.

Let S be the event of picking a Spade. P(S) is the probability of picking a Spade.
We can rewrite the multiplication rule:
P(O and S) = P(O) × P(S)
3
Find the probability of rolling an odd number on a die.
There are 3 ways of rolling an odd number on a die...
...out of 6 possible outomes from rolling the die: getting 1, 2, 3, 4, 5 and 6.
The probability of rolling an odd number on a die is ^{3}⁄_{6}. P(O) = ^{3}⁄_{6}.
4
Find the probability of picking a Spade from a deck of cards.
There are 13 ways of picking a Spade from a deck of 52 cards.
The probability of picking a Spade from a deck of cards is ^{13}⁄_{52}. P(S) = ^{13}⁄_{52}.
5
Substitute the probability of rolling an odd number and the probability of picking a Spade into the formula.
P(O and S) = ^{3}⁄_{6} × ^{13}⁄_{52}
P(O and S) = ^{39}⁄_{312}
6
Simplify the fraction if possible.
Simplify the fraction by dividing the top and bottom numbers by their greatest common factor (which is 39).
39 ÷ 39 = 1
312 ÷ 39 = 8
Answer:
The probability of rolling an odd number on a die and picking a Spade from a deck of cards is ^{1}⁄_{8}.
We have expressed the probability as a fraction. We can also express this as a number (0.125) or a percentage (12.5%).