1. Tree diagrams
Tree diagrams are used to display the sample space for the experiment or game. The tree branches out once for every stage of the experiment.
Below are the tree diagrams for flipping a coin and rolling a die and flipping three coins.
Each branch represents a new event such as each flip of a coin or each child a mother has.
Coin 3
Coin 2
Die
Coin 1 H T
HHH
Coin H1 H T 1 2 3 4 5 6 H2
HHT H3 H4 H5 H6 H T O H T O H T
HTH
HTT
THH
THT
TTH
TTT H T 1 2 3 4 5 6 T1 T2 T3 T4 T5 T6 H T H T

2. Probability tree diagrams
With probability tree diagrams you place the probability of each event on each arm of the diagram.
If there is no replacement in a question the probabilities on the arms change after each event.
2 D’s and 8 N’s are placed in a hat.
3 “letters” are drawn WITHOUT replacement.
Draw a probability tree to show the outcomes.
Example 1
Note the probabilities on each set of arms adds to 1
DON’T simplify the probabilities. This means all answers will have the same denominators

3. Example 2 A hat has 5 red disks and 3 black disks.
Two disks are drawn out without replacement.
What is the probability of drawing:
different coloured disks?
2 red disks?
a) P(different) =
15/56
+ 15/56
= 30/56
= 15/28
Note the probabilities on each set of arms adds to 1
Disk 2
Disk 1
4/7
20/56 R B RR  
b) P(RR) =
20/56 R B
= 5/14
5/8
3/7 RB
15/56  
5/7 R B BR
15/56  
3/8
DON’T simplify the probabilities. This means all answers will have the same denominators
2/7
6/56 BB  
56/56
Check

4. Example 3
The probability of a Sydney resident catching the flu is 60%.
In a sample of 3 Sydney residents, what is the probability that;
exactly 2 residents have the flu?
at least one resident has the flu?
The resident can be either have the flu (F, 3/5) or be healthy (H , 2/5).
Resident 3
As the population is so large, the probability does not change.
Resident 2
3/5
27/125
3/5 F H
FFF F H
Resident 1
2/5
FFH
18/125
a) P(2F, 1H) =
18/125
× 3 F H
3/5
3/5 F H
FHF
18/125
= 54/125
2/5
2/5
FHH
12/125
b) P(≥1F) =
1 − P(3H)
3/5
3/5 F H
HFF
18/125
= 1 − 8/125
= 117/125 F H
2/5
2/5
HFH
12/125
3/5 F H
HHF
12/125
2/5
2/5
HHH
8/125
Check
125/125

5. Today’s work Yesterday’s work Exercise 6-07 Page 219 → 220
1 → 4, 7, 9 → 11 & 14
Yesterday’s work
Exercise 6-06
Page 214 → 216
Odds