The Multiplication Rule of Probability(KS3, Year 7)

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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 12. It tells us that when a die is rolled, the probability of rolling a 6 is 16. 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.

Question

What is the probability of getting Heads in a coin toss and rolling a 6 on a die?

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 12.

4

Find the probability of rolling a 6. The probability of rolling a 6 is 16.

5

Substitute the probability of tossing Heads (12) and rolling a 6 (16) into the formula.
12 × 16 = 112

The probability of tossing Heads and rolling a 6 is 112.

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?

1

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 36. P(O) = 36.

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 1352. P(S) = 1352.

5

Substitute the probability of rolling an odd number and the probability of picking a Spade into the formula.

P(O and S) = 36 × 1352

P(O and S) = 39312

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

The probability of rolling an odd number on a die and picking a Spade from a deck of cards is 18. We have expressed the probability as a fraction. We can also express this as a number (0.125) or a percentage (12.5%).

Lesson Slides

The slider below another real example of using the multiplication rule of probability. In this example, the probability of three events is found using the multiplication rule.

And = ×

P(A and B) = P(A) × P(B)
and = ×

The Multiplication Rule Is for Independent Events

The multiplication rule works for independent events not dependent events.

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)

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