Unit 10 Section 2 : Finding Outcomes With Two Experiments

When two experiments take place at the same time, it is often useful to list all the possible outcomes which can happen.
There are three possible approaches: writing out a list, using a table or using a tree diagram.

Example Questions

In the example questions below, read the explanations, then work out the answers which are not shown.
You can then click on the button marked Click on this button below to see the correct answer to see whether you are correct.

Example Question 1
Caitlin and Dave each buy a chocolate bar from a vending machine that sells Aero, Bounty, Crunchie and Dime bars.
List the possible pairs of bars which Caitlin and Dave can choose.
To answer this systematically, start by assuming that Caitlin has chosen an Aero, and then look at what Dave can choose - he can choose any of the four types of chocolate bar, so we get this list:

So far there are four outcomes, but Caitlin can choose any of the three other chocolate bars as well. Repeating the process with each bar Caitlin can choose gives us the table on the right, which shows all the outcomes.

Note that Aero/Bounty and Bounty/Aero are different outcomes, as the bars have been chosen by different people.

How many possible outcomes are there in this situation?

Example Question 2
An unbiased coin is tossed and a fair dice is rolled. Draw a table to show the possible outcomes.

This type of table is sometimes called a possibility space, because it shows all the possibilities in a situation. The possible outcomes for the first experiment (the coin) are shown on the left, and the possible outcomes for the second experiment (the dice) are shown on the top.
In the middle of the table we write the combination of the two outcomes together.
For example, "H1" means a head "H" on the coin and a number "1" on the dice.

How many possible outcomes are there in this situation?

Example Question 3
Draw a table to show all the possible outcomes which occur when you add together the scores on two fair dice.

This time we put the outcomes of the first dice on the left, and the outcomes of the second dice on the top.
In the middle of the table we put the total we get when we add the scores together.
For example, the top left box is a "2" because 1 + 1 = 2.
Note that the three outcomes marked "4" are counted as different outcomes, even though they give the same value, because they have been achieved in different ways: 1 + 3 = 4, 2 + 2 = 4 and 3 + 1 = 4.

(a) What value should go where the red question mark is?

(b) How many possible outcomes are there in total?

(c) How many outcomes give a score of 8?

Example Question 4
Use a tree diagram to show the possible outcomes when two unbiased coins are tossed.

Although we could use systematic listing or a table to find the outcomes in this situation, we can also do it with a tree diagram. Possibilities are represented by branches on a tree. You start by considering the possible outcomes on the first coin:
Regardless of whether the first was heads (H) or tails (T), there are still two possibilities (H and T) for the second coin:
The final outcomes are found by following all possible routes along the branches:
Note that HT and TH are different outcomes even though they both consist of one tail and one head.

How many possible outcomes are there in this situation?

Example Question 5
Tim has two black socks, two white socks and two red socks in a drawer. He pulls out two socks.
Draw a tree diagram to find the possible outcomes.

For the first sock, there are three possible outcomes. For the second sock, there are the same three outcomes.
The tree diagram will look like this:
We can see that there are nine outcomes in this situation.

How many outcomes are there where both socks are the same colour?

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct 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 Click on this button to see the correct answer to see the answer.

Question 1
Two fair coins are tossed.

(a) Complete the following table.
Write the outcomes in capital letters with no spaces like this .
Make sure the letters are in the correct order (COIN 1 then COIN 2).
(b) How many possible outcomes are there in this situation?

Question 2
Two spinners are numbered 1 to 4 as shown in this diagram:

The spinners are both spun and the scores are added together to give the outcome.

(a) Complete the table below, to show all possible outcomes.
(b) What is the total number of possible outcomes?

(c) How many outcomes give a score of 5?

Question 3
Two spinners are numbered as shown in this diagram:

The spinners are both spun and the scores are multiplied together to give the outcome.

(a) Complete the table below, to show all possible outcomes.
(b) How many outcomes are there?

(c) How many outcomes give a score of 6 or more?

Question 4
In a bag there are red (R) and blue (B) counters. Two counters are taken out of the bag at random.

(a) Complete the tree diagram below, to show the possibilities on the branches and the final outcomes.
Write the outcomes in capital letters with no spaces. Make sure the outcome letters are in the correct order.
(b) How many outcomes include a red counter?

(c) How many outcomes include a blue counter?

Question 5
In a jar there are three different types of sweets, chews (C), mints (M) and toffees (T). Two sweets are taken at random.

(a) Fill in the gaps on the tree diagram below.
Write the outcomes in capital letters with no spaces. Make sure the outcome letters are in the correct order.
(b) How many of the outcomes include a toffee?

(c) How many of the outcomes include a mint and a toffee?

You will need some paper and something to write with for the next three questions.
You should choose one of the methods detailed above to work out possible outcomes.
You can then type in the answers to each question in the boxes below and check them.
Question 6
A red dice, a blue dice and a green dice are put into a bag; all the dice are fair.
One is then taken out and rolled. The colour of the dice and the score shown are recorded.

(a) How many possible outcomes are there?

(b) How many outcomes include a 5?

Question 7
In a game, two fair dice are rolled and the scores are multiplied together.

(a) How many possible outcomes are there?

(b) How many ways are there of scoring 12?

(c) How many ways are there of scoring 18?

Question 8
A bag contains a mixture of red, green and white balls. Three balls are taken at random from the bag.

(a) How many possible outcomes are there? (HINT: Use a big tree diagram or write out a long list)

(b) How many outcomes include a red ball?

(c) How many outcomes include a red or a white ball?

(d) How many outcomes include a red and a green ball?


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Produced by A.J. Reynolds August 2007