Unit 10 Section 3 : Finding Probabilities Using Lists Of Outcomes

In Unit 10 Section 2 we saw three methods of listing all the outcomes when two or more experiments take place at the same time. If the outcomes in each of these experiments are equally likely, then the combined outcomes will also be equally likely.

Example Question

When two unbiased coins are tossed, determine the probability of obtaining:
(a) two heads,
(b) two tails,
(c) a head and a tail.

One of the ways we can list all the possible outcomes is to use a table (or possibility space).
We can see that there are four possible outcomes in this situation: HH, HT, TH and TT. Because the outcomes on each coin are equally likely, the four "combined" outcomes are also equally likely. This means we can now use this formula to work out the probability of particular events:
(a) Probability of "two heads" = number of successful outcomes (those which have two heads)  =  1
number of possible outcomes (from both experiments) 4

(b) Probability of "two tails" = number of successful outcomes (those which have two tails)  =  1
number of possible outcomes (from both experiments) 4

(c) Probability of "a head and a tail" = number of successful outcomes (those with a head and a tail)  =  2  =  1
number of possible outcomes (from both experiments) 4 2

Practice Questions

Work out the answer to each of these questions then click on the button marked Click on this button below to see the correct answer to see whether you are correct.

Practice Question 1
Caitlin and Dave each buy a chocolate bar from a vending machine that sells Aero, Bounty, Crunchie and Dime bars. What is the probability that they both choose the same type of chocolate bar?.

In the previous section we systematically listed all the possible outcomes in this situation. The list of possible outcomes is shown on the right.

Because Caitlin and Dave are choosing randomly, each chocolate bar is equally likely to be chosen. Each of the combined outcomes shown on the right is also therefore equally likely.

(a) In how many outcomes did Caitlin and Dave choose the same chocolate bar?

(b) How many possible outcomes are there in this situation?

(c) What is the probability that they choose the same type of chocolate bar?

Practice Question 2
Two fair dice are rolled at the same time and the scores are added together.

A table can be used to find the 36 possible outcomes.
The outcomes on each dice are equally likely, so these 36 combined outcomes will also be equally likely.

(a) What is the probability of obtaining a score of 6?

(b) What is the probability of obtaining a score greater than 9?

(c) What is the probability of obtaining a score less than 7?

 

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
The diagram below shows two spinners which are both spun. The scores are then added together.
You might find it helpful to fill in the table below.
The table will not be marked.

Don't forget to cancel your fraction answers down.

What is the probability that:
(a) the total score is 7?   

(b) the total score is greater than 10?   

(c) the total score is 6?   

(d) the total score is less than 5?   

Question 2
An unbiased coin is tossed and a fair dice is thrown.
Use this table of outcomes to answer the questions below.

(a) What is the probability of obtaining a head and a 3?

(b) What is the probability of obtaining a tail and an even number?

(c) What is the probability of obtaining a head and a square number?

Question 3
The diagram below shows two spinners which are both spun. The scores are then added together.
You might find it helpful to fill in the table below.
The table will not be marked.

Don't forget to cancel your fraction answers down.

What is the probability that the total score is:
(a) six?   
(b) zero?   
(c) one?   
(d) three?   
(e) an even number?   
(f) greater than one?   
(g) less than one?   
(h) less than six?   

You will need some paper and something to write with for the next two questions.
You should use the method specified in each question to work out possible outcomes.
You can then type in the answers to each question in the boxes below and check them.
Question 4
Three unbiased coins are tossed at the same time.
Use a tree diagram to work out the possible outcomes.

What is the probability of obtaining:
(a) 3 heads?

(b) at least 1 head?

(c) at least 2 heads?

Question 5
Two fair dice are rolled and the scores on each dice are multiplied together to give a total score.
Use a table to work out all the possible outcomes.

What is the probability of getting a total score:
(a) of 12?

(b) of 20?

(c) greater than 25?

(d) less than 30?

(e) that is an even number?


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Produced by A.J. Reynolds August 2007