Introduction to Probability
Do you think there is a problem?
Consider the map of counties shown below. The number in each
county is last month’s incidence rate for a disease in cases per 100,000
population.
STATISTICAL VERSUS MATHEMATICAL
THINKING
Mathematical Thinking
Explain Patterns
Often Deterministic
Statistical Thinking
Search for Patterns in the presence of variability
Acknowledge role of variation
Statisticians like to ask “Could this have happened by chance?”
Cookie Game
Cookie Game
• Statistical Thinking
• Could this have happened by chance?
• Convincing evidence versus proof
• Acknowledging the risk of an incorrect conclusion
• How do we measure likelihood?
• How do we measure risk versus reward?
Simple Definitions
• An experiment is any process that allows researchers to obtain
observations
• The sample space of an experiment is the collection of all possible
outcomes
• An event is any subset of the sample space
Example
• The experiment is rolling a single die
• The sample space: S = {1, 2, 3, 4, 5, 6} since you can get only 1
through 6 as outcomes when you roll a die
• Here are two events:
Rolling a 3
Rolling an odd number - this event is {1, 3, 5}
Example
• The experiment is drawing one card from a standard deck of cards
and noting the suit of the card
• The sample space: S={club, heart, spades, diamonds}
• One event is getting a red card which we can represent as {heart,
diamond}
• Another event is getting a spade
Roulette
Calculating Probabilities
Theoretical (Classical) approach
Assume that the experiment has n different outcomes, each of
which have an equal chance of occurring. If the event A can
occur in s of these ways then
Probability of A = P(A) = (number of ways A can occur)/(number
of total outcomes)
The notation P(A) is read probability of A
Probability Facts
•0≤𝑃 𝐴 ≤1
• 𝑃(𝑁𝑂𝑇 𝐴) = 1 − 𝑃(𝐴)
Roll two dice
Here is the sample space of equally likely outcomes
Yellow Die
1
2
3
4
5
6
Red Die
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Empirical (experimental) Probability
Sometimes it is not possible to construct a sample space in a manner
where the outcomes are equally likely. In this case we conduct an
experiment n time (these are called trials) We count the number of
times each outcome occurs and this gives us an estimate of our
probability
number of times A occurs
• 𝑃(𝐴) ≈
Total number of trials
Law of Large Numbers
Know the Law
If you conduct an experiment a large number times, the
experimental probability will approach the actual probability