1. Theoretical Probability
Section 2

2. Two types of probabilities
Empirical Probability
Theoretical Probability
Empirical probability tells us the frequency that an event happened during repeated trials of an experiment.
Theoretical probability allows us to predict the frequency of those repeated trials.

3. Theoretical Probability
We assume that each outcome of an experiment happens with equal chance.
The theoretical probability of an event E occuring in an experiment is
Throughout the course, we’ll refer to theoretical probability as just probability.

4. Example
There are six different outcomes of the experiment “rolling a six-sided die”: 1, 2, 3, 4, 5 and 6.
Let E be the event of rolling a 2. The probability of E is

5. All possible outcomes of “rolling a die” experiment
The event of rolling a 2

6. Sample space
The sample space of an experiment is the set of all of the outcomes of that experiment.
For example, the sample space of the “rolling a die” experiment is the set {1, 2, 3, 4, 5, 6}.
We can then think of events as subsets of the sample space.

7. P(E) = # outcomes in E / # outcomes in sample space
Event E
Sample space

8. Sample space of “roll a die”
Event of rolling a 2
Sample space

9. More examples with “roll a die”
B (empty set) C
Events
A = “a number greater than 4”
B = “7”
C = “a number less than 7”
P(A) = 2/6 = 1/3
P(B) = 0/6 = 0
P(C) = 6/6 = 1
Sample space A

10. Complements
The complement of an event E is the event within the same sample space as E in which E does not occur.
We denote it with EC. Also called “not E”.
We have the following two identities between E and EC:

11. Complements
Event EC
Event E
Sample space

12. Complements in “roll a die” sample space
DC (empty set)
BC (“a number less than or equal to 4”) AC
Events
A = “2”
B = “a number greater than 4”
D = “a number less than 7”
Sample space

13. Both are equally likely!
Question
Suppose we flip a coin 10 times. Which of the following is more likely to happen?
HHHTHHTTTH Or
HHHHHHHHHH
Both are equally likely!

14. “Number less than or equal to 3” = {1, 2, 3}
For the “rolling a six-sided die” experiment, what is the probability of rolling either an even number or rolling a number less than or equal to 3?
“Even number” = {2, 4, 6}
“Number less than or equal to 3” = {1, 2, 3}
“Even number OR number less than or equal to 3” = {1, 2, 3, 4, 6}
Probability = 5/6

15. Sample space of “roll a die”
“Even number”
“Number less than or equal to 3”
Sample space

16. P(“even number OR number ≤ 3”)
= P(“even number”) + P(“number ≤ 3”) - P(“even number AND number ≤ 3”)
= ½ + ½ - ⅙
= ⅚
In general,