1. 27/11/2018
Tree Diagrams

2. 27/11/2018
Tree diagrams
Help you to solve probability problems where one thing is happening after another (after another…)
For example
picking two or more items out from a selection
playing two or more different games
considering whether something will happen on two or more different days, etc.
You can use fractions or decimals to show the probabilities in a tree diagram
We are going to look at INDEPENDENT events first

3. The following situation can be represented by a tree diagram.
27/11/2018
The following situation can be represented by a tree diagram.
Peter has ten coloured cubes in a bag. Three of the cubes are red and seven are blue. He removes a cube at random from the bag and notes the colour before replacing it. He then chooses a second cube at random. Record the information in a tree diagram.
Second Choice
First Choice
red
red
blue
red
blue
blue

4. 27/11/2018 Characteristics of a tree diagram… red blue Second Choice
First Choice
Second Choice
The probabilities for each event are shown along the arm of each branch and they sum to 1.
Ends of first and second level branches show the different outcomes.
Probabilities are multiplied along each arm.

5. 27/11/2018 black black green green
Question 1 Rebecca has nine coloured beads in a bag. Four of the beads are black and the rest are green. She removes a bead at random from the bag and notes the colour before replacing it. She then chooses a second bead. (a) Draw a tree diagram showing all possible outcomes. (b) Calculate the probability that Rebecca chooses: (i) 2 green beads (ii) A black followed by a green bead.
Second Choice
black
green
black
green
First Choice

6. 27/11/2018 Peter Win Peter Win Becky Win Peter Win Becky Win Becky Win
Question 3 Peter and Becky run a race and play a tennis match. The probability that Peter wins the race is 0.4. The probability that Becky wins the tennis is 0.7.
Complete the tree diagram below.
Use your tree diagram to calculate (i) the probability that Peter wins both events (ii) The probability that the race and the game are won by different people
Tennis
Race
0.3
Peter Win
0.4 x 0.3 = 0.12
Peter Win
0.4
Becky Win
0.7
0.4 x 0.7 = 0.28
0.3
Peter Win
0.6 x 0.3 = 0.18
0.6
Becky Win
0.7
Becky Win

7. 27/11/2018 1. Simon plays one game of tennis and one game of snooker.
The probability that Simon will win at snooker is
The probability that Simon will win at tennis is
a) Complete the tree diagram
b) Work out the probability that Simon wins both games.
c) Work out the probability that Simon will win only one game.

8. 27/11/2018
Example

9. 27/11/2018
Example

10. 27/11/2018
Now have a go at the booklet questions

11. 27/11/2018
NON-REPLACEMENT

12. 27/11/2018
Non-replacement questions
Sometimes you will not want to replace the item before making the second selection
For example
Choosing items of food or drink – you will not be putting it back as it will be consumed!
Choosing a pair of socks – once you choose the first sock you will keep it to wear, not put it back
Choosing two people from a group to help you with a job – once the first person is chosen the group you are selecting from the second time is now different
These questions are about DEPENDENT events

13. 27/11/2018 The light bulb in your room has blown. You are in a rush.
There are twenty socks in a drawer. Fourteen of them are blue and the rest are green.
1st sock
2nd sock
Blue 13 19
Blue 6 19 14 20
Green 6 20
Blue 14 19
Green 5 19
Green

14. 27/11/2018 Example A bag contains 3 red and 7 blue balls.
A ball is picked, not replaced, and another picked.
Complete the tree diagram
There are 10 balls to choose from when picking the 1st ball
1st pick
2nd pick
If the object is not replaced, this affects the probabilities on the 2nd pick
Red
If a red ball was picked first, there are only 2 red balls left
Red
Blue
If a red ball was picked first, there are still 7 blue balls left
If a blue ball was picked first, there are still 3 red balls left
Red
Blue
If a blue ball was picked first, there are only 6 blue balls left
Blue

15. 27/11/2018
Frequency Trees

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These do not show probabilities but instead show actual numbers. Can you complete the tree diagram below showing the number of people selecting options for lunch from a sample of 75 28
How many people chose wedges?
Chips 41
Nuggets
Wedges
If a person was selected at random what is the probability they had nuggets and wedges? 25
Chips
Burger
Wedges

17. 27/11/2018
Can you complete the tree diagram below showing the number of people passing two tests from a sample of 22. Half of the people passed both tests.
How many people passed 1 test exactly
Passed
Passed 4
Failed
If a person was selected at random what is the probability they failed both tests? 5
Passed
Failed
Failed