# The 'And' Rule of Probability (Mathematics Lesson)

# What Is the 'And' Rule of Probability?

Probability tells us how likely something is to happen.It can be used to find out how likely it is to get a Head when tossing a coin:

Or how likely it is to roll a 6 on a die:

But we might be interested in finding out the probability that one event

**and**another event occurs.

For example,

- What is the probability of tossing a Head
**and**a Tail? - What is the probability of rolling a 6
**and**then another 6 on a die?

# How to Use the 'And' Rule of Probability

For two independent events,**A**and

**B**, the probability of

**A and B**both occuring is given by multiplying the probability of

**A**occuring with the probability of

**B**occuring:

# A Real Example of How to Use the 'And' Rule of Probability

**Question:**What is the probability of getting a Head

**and**then a Tail in two coin tosses?

**Step 1:**Find the probability of one event.

The probability of getting a Head, P(H), is:

**Step 2:**Find the probability of the other event.

The probability of getting a Tail, P(T), is:

**Step 3:**Multiply the probabilities together.

The probability of getting a Head and then a Tail is ¼.

# How to Use the 'And' Rule of Probability from a Tree Diagram

The tree diagram below represents two tossings of a coin.Find the branches that represent getting a

**H**ead in the 1

^{st}toss and a

**T**ail in the 2

^{nd}toss, and multiply the probabilities along the branches together:

This confirms that the probability of getting a Head and then a Tail is ¼.

# Another Real Example of How to Use the 'And' Rule of Probability

The slider below gives a real example of how to use the 'and' rule to find the probability that two independent events both occur.##### Interactive Test

**show**

##### Top Tip

**PROBABILITY OF 'AND' = ×**

For independent events, A and B,

P(A

**and**B) = P(A) × P(B)

**and**= ×

**THIS DOESN'T WORK FOR DEPENDENT EVENTS**

The method discussed here only works for independent events, not for dependent events.

##### Note

# MULTIPLYING FRACTIONS

As probabilities are often written as fractions, mutliplying them involves multiplying fractions.To multiply fractions, e.g.:

- Multiply the numerators together:

- Multiply the denominators together:

**WHAT IS PROBABILITY?**

Probability tells us how likely something is to happen.

The probability of an event is defined as:

Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible. The probability of pigs flying is 0 (unless they are put in an aeroplane!).

A probability of 1 means an event is certain. The probability that an elephant is heavier than a fly is 1.

A probability of ½, such as getting a Heads in a coin toss, means it is equally likely to go either way. A coin is as likely to come up Heads as it is to come up Tails.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.

**WHAT IS AN INDEPENDENT EVENT?**

Two events are independent if the probability of one event occuring does not depend on whether the other event occurs.

For example, if I toss a coin, there is always a ½ probability of it coming up Heads, regardless of how many times I toss it.