Unit 10 Section 4 : Multiplication Law For Independent Events
If a particular outcome or event in one experiment does not affect the probabililty of outcomes or events in a second experiment, then the
outcomes or events are said to be independent.
For example, if you roll a dice twice, then the outcome in the first roll will not affect the probabilities of the outcomes when the dice is rolled a second time.
The results of the two dice rolls are independent of each other.
Multiplication Law
If X and Y are two independent events with probabilities P(X) and P(Y) respectively, then the probability that X and Y will both happen
is found by multiplying the two probabilities together:
With this we can calculate probabilities of combined outcomes when the original experiment outcomes are not equallylikely.
Example Questions
Example Question 1
Two fair dice are rolled. Use a probability tree diagram to determine the probability of obtaining:
(a) two sixes,
(b) no sixes,
(c) exactly one six.
We are only interested if the dice shows a six or not, so we have two outcomes, "six" and "not six".
The probability of "six" is and the probability of "not six" is .
We start by drawing a tree diagram to show the outcomes:
Then we write on the probability for each branch.
Each set of branches should add up to 1 (e.g. + = 1).
Finally, we work out the probability of each outcome by multiplying the probabilities along the branches which lead to that outcome.
For example, the probability P(six AND not six) is found by multiplying P(six) by P(not six).
Note how all the outcome probabilities also add up to 1  this will always happen and is a useful thing to check.
So, the answers are as follows:
(a) P( two sixes ) = P( six , six ) =
(b) P( no sixes ) = P( not six , not six ) =

To find the probability of "exactly one six", we need to combine the outcomes for ( six , not six ) and ( not six , six ), as they both have "exactly one six".
We combine these outcomes by adding their probabilities:
(c) P( exactly one six ) = P ( six , not six ) + P( not six , six ) =

Example Question 2
Russell is playing in a cricket match and a game of football at the weekend.
The probability that his team will win the cricket match is 0.7, and the probability of winning is 0.9 in the football.
Assume that the results in the matches are independent  they do not affect each other.
What is the probability that his team will win in both matches?
Although we could draw a probability tree in this situation, we do not need to.
We can simply use the multiplication law:
P(win both matches) = P(win cricket AND win football) = P(win cricket) × P(win football) = 0.7 × 0.9 = 0.63
While a probability tree is often useful, it is worth checking whether simpler questions can be answered without one.
Practice Questions
Work out the answer to each of these questions then click on the button marked
to see whether you are correct.
Practice Question 1
A bag contains 4 red balls and 3 green balls.
A ball is taken at random, and then put back. A second ball is then taken from the bag.
The results are independent because the first ball is put back before the second is taken out.
(a) Draw a probability tree, showing the possible outcomes and their probabilities.
(b) What is the probability that:
(i) both balls are the same colour 

(ii) the balls are of different colours 

(iii) at least one of the balls is green 

Practice Question 2
On her way to work, Sylvia drives through three sets of traffic lights.
The probability of each set of lights being green is 0.3.
What is the probability that they are all green?
Exercises
Work out the answers to the questions below and fill in the boxes. Click on the
button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next
question. If appears then your answer is wrong. Click
on to clear your original answer and have another go.
If you can't work out the right answer then click on
to see
the answer.
You will need some paper and something to write with for the next few questions.
You should draw a tree diagram if necessary in each question to work out the probabilities.
You can then type in the answers to each question in the boxes and check them.


You have now completed Unit 10 Section 4
Since these pages are under
development, we find any feedback extremely valuable. Click
here to fill out a very short form which allows you make comments about
the page, or simply confirm that everything works correctly.

Return to the Y8 Tutorials Menu
Produced by A.J. Reynolds August 2007
