1. Probability – the bedrock of randomness Definitions Random experiment – observing the close of the NYSE and the Nasdaq Sample space = {NYSE+Nasdaq+, NYSE+Nasdaq-, NYSE-Nasdaq+, NYSE- Nasdaq-} Simple event = {NYSE+Nasdaq+} Event = the NYSE is up

2. Approaches to probability Classic – equal likelihood Dice, cards, inventory control, polls, audits – (random samples) Relative frequency – historic data, relative frequency distributions Actuarial tables Race track odds Subjective Which for the NYSE and the Nasdaq?

3. Probabilities of Combinations of Events Union of sets – “or” The NYSE is up or Nasdaq is up Intersection of sets – “and” Joint events The NYSE is up and the Nasdaq is up

4. Conditional Probability P (A|B) = P(A and B)/P(B) A and B are independent if P(A|B) = P(A) Which also means that P(B|A) = P(B) Probability the Nasdaq is up given that the NYSE is up.

5. Rules of Probability Complement Addition General Special Multiplication General Special

6. Rules: Complement: P(A complement) = 1 – P(A) Addition: P(A or B) = P(A) + P(B) – P(A and B) P(A or B) = P(A) + P(B) if A and B are mutually exclusive Multiplication:P(A and B) = P(A|B) * P(B) P(B|A) * P(A) P(A and B) = P(A) * P(B) if A and B are independent

7. Probability table

9. Randomness and Probability ‘Scientific sampling’ is random sampling Simple Stratified Cluster What? Why? How?

10. What is random sampling? Simple random sample -Every sample with the same number of observations has the same probability of being chosen Stratified random sample – Choose simple random samples from the mutually exclusive strata of a population Cluster sample – Choose a simple random sample of groups or clusters

11. Why sample randomly? To make valid statistical inferences to a population Conclusions from a convenience sample can be questioned Conclusions from a self-selected sample are SLOP

12. How can samples be randomly chosen? Random number generators (software) Ping pong balls in a hopper Other mechanical devices Random number tables Slips of paper in a ‘hat’

13. Summary: Only inferences from random samples are valid The approach to assigning probabilities must be chosen: any probability must be between 0 and 1, inclusive the probability of the sample space is 1 The language of probability is the language of set theory. Learn the complement, addition and multiplication rules. Tables help in determining joint and conditional probabilities.