1. Using Tree Diagrams to Represent a Sample Space and to Calculate Probabilities

2. Probability Tree Diagrams
Let’s look at the probability of a coin toss
There are two "branches" (Heads and Tails)
The probability of each branch is written on the branch
The outcome is written at the end of the branch

3. Probability of 2 coin tosses
We can extend the tree diagram to two tosses of a coin

4. How do you calculate the probability?
2 STEPS!
You multiply probabilities along the branches
You add probabilities down columns
(to make sure it equals 1)

5. Example

6. Example: Soccer Game
You are off to soccer, and love being the Goalkeeper, but that depends who is the Coach today:
with Coach Sam the probability of being Goalkeeper is 0.5
with Coach Alex the probability of being Goalkeeper is 0.3
How would you start your tree diagram?

7. Probability Tree Diagram
Sam is Coach more often ... about 6 out of every 10 games (a probability of 0.6).
The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)

8. Now, if you get Sam, there is 0. 5 probability of being Goalie (and 0
Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):

9. If you get Alex, there is 0. 3 probability of being Goalie (and 0
If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):

10. The tree diagram is complete, now let's calculate the overall probabilities. This is done by multiplying each probability along the "branches" of the tree.
Here is how to do it for the "Sam, Yes" branch:

11. Will you be goalie today?
An 0.4 chance of Alex as Coach, followed by an 0.3 chance gives 0.12.
= 0.42 probability of being a Goalkeeper today

12. Example 2 Taylor decides to roll a die 2 times.
Create a tree diagram to show the probability of her landing on 2 both times.

13. Births
Draw a tree diagram demonstrating the four possible birth outcomes for a family with 𝟐 children (no twins). Use the symbol “𝑩” for the outcome of “boy” and “G” for the outcome of “girl.” Consider the first birth to be the “first stage.”