The Lesson
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?Step-by-Step:
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
4
Find the probability of rolling a 6.
The probability of rolling a 6 is ^{1}⁄_{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?Step-by-Step:
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.
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}.
...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}.
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%).
Top Tip
And = ×
P(A and B) = P(A) × P(B)
and = ×
Beware
The Multiplication Rule Is for Independent Events
The multiplication rule works for independent events not dependent events.Note
A Note on Notation
The probability of an event can be written as:
P(Event)
A letter or symbol can be used to represent an event.
For example, let H be the event that a coin lands on Heads when it has been tossed.
We can denote the probability of getting heads as:
P(H)