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 Head in the 1st toss and a Tail in the 2nd 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.
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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.

MULTIPLYING FRACTIONS

As probabilities are often written as fractions, mutliplying them involves multiplying fractions.

To multiply fractions, e.g.:

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.