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Probability – the bedrock of randomness

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. Approaches to probability.

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Probability – the bedrock of randomness

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  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 theNasdaq is up

  4. Conditional Probability P (A|B) = P(A and B)/P(B) A and B are independentif 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

  8. 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.

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