*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

**. It tells us that when a die is rolled, the probability of rolling a**

^{1}⁄_{2}**6**is

**. Imagine we wanted to find the probability of tossing**

^{1}⁄_{6}*Heads*

**and**rolling a

*6*.

## The Multiplication Rule

To find a probability of one event**and**another event...

Probability of Heads

...replace the **and**Probability of 6**and**with a

**×**...

Probability of Heads

This is the multiplication rule.
**×**Probability of 6## 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}## 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**event

__and__**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

...out of

The probability of rolling an

**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

The probability of picking a

**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%).

## 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)*

**H**be the event that a coin lands on

**H**eads when it has been tossed. We can denote the probability of getting heads as:

*P(H)*

## You might also like...

probabilityunderstanding independent eventsunderstanding dependent eventsunderstanding mutually exclusive events

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