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6 October 2003

6 October 2003. 4.1 Randomness What does “random” mean? 4.2 Probability Models 4.5 General Probability Rules Defining random processes mathematically Combining probabilities: The Addition Rule The Multiplication Rule Conditional probabilities Decision analysis. RANDOMNESS.

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6 October 2003

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  1. 6 October 2003 • 4.1 Randomness • What does “random” mean? • 4.2 Probability Models • 4.5 General Probability Rules • Defining random processes mathematically • Combining probabilities: • The Addition Rule • The Multiplication Rule • Conditional probabilities • Decision analysis

  2. RANDOMNESS • Random is not the same as haphazard or helter-skelter or higgledy-piggledy. • Random events are unpredictable in the short-term, but lawful and well behaved in the long-run. • For example, if I toss one coin, I do not know whether it will land heads or tails. But if I toss a million coins, I can be reasonably certain that about half of them will be heads and the other half tails.

  3. PROBABILITY • Probabilities are numbers which describe the outcomes of random events. • The probability of an event is the long-run relative frequency of that event. • P(A) means “the probability of event A.” • If A is certain, then P(A) = one • If A is impossible, then P(A) = zero

  4. Sample Space • A “sample space” is a list of all possible outcomes of a random process. • When I roll a die, the sample space is {1, 2, 3, 4, 5, 6}. • When I toss a coin, the sample space is {head, tail}. • An “event” is one or more members of the sample space. • For example, “head” is a possible event when I toss a coin. Or “number less than four” is a possible event when I roll a die.

  5. Probability Rules • All probabilities are between zero and one: • 0 < P(A) < 1 • Something has to happen: • P(Sample space) = 1 • The probability that something happens is one minus the probability that it doesn’t: • P(A) = 1 - P(not A)

  6. Examples • The probability that I wear a green shirt tomorrow is some number between zero and one. • 0 < P(green shirt) < 1 • The probability that I wear a shirt of some color tomorrow is equal to one. • P(shirt) = 1 • The probability that I wear a green shirt tomorrow is one minus the probability that I don’t wear one. • P(green shirt) = 1 - P(non-green shirt)

  7. CHANCES and ODDS • Chances are probabilities expressed as percents. Chances range from 0% to 100%. • For example, a probability of .75 is the same as a 75% chance. • The odds for an event is the probability that the event happens, divided by the probability that the event doesn’t happen. Odds can be any positive number. • For example, a probability of .75 is the same as 3-to-1 odds.

  8. Conditional Probability • The conditional probability of B, given A, is written as P(B|A). It is the probability of event B, given that A occurs. For example, P(blue pants | green shirt) is the probability that I will put on a pair of blue pants, given that I have already picked out a green shirt. • Note that P(B|A) is not the same as P(A|B).

  9. Independence • Events A and B are independent if the probabiity of event B is not affected by A’s occurring or not occurring: • If and only if A and B are independent, P(B | A) = P(B | not A) = P(B) • For example, if I am tossing two coins, the probability that the second coin lands heads is always .50, whether or not the first coin lands heads. • P(H2 | H1) = P(H2|T1) = P(H2)

  10. Non-independence • Events A and B are not independent if P(B) is different, depending on whether A occurs: • If P(B | A) ≠ P(B | not A), then A and B are not independent. • Suppose I don’t like to wear blue pants with a green shirt: • P(blue pants|green shirt) < P(blue pants|not-green shirt). • “Blue pants” and “green shirt” are not independent.

  11. The Addition Rule • If A and B cannot both occur, then • P(A or B) = P(A) + P(B) • P(green shirt or blue shirt) = P(green shirt) + P(blue shirt) • The events “green shirt” and “blue shirt” are called disjoint. • If A and B could both occur, then • P(A or B) = P(A) + P(B) - P(A and B) • P(green shirt or blue pants) • = P(green shirt) + P(blue pants) - P(green shirt and blue pants) • The probability that I wear green shirt or blue pants is the probability that I wear a green shirt PLUS the probability that I wear blue pants MINUS the probability that I wear a green shirt and blue pants.

  12. The Multiplication Rule • If A and B are independent, then • P(A and B) = P(A) x P(B) • For example, if I choose my shirts and pants separately, then: • P(green shirt and blue pants) = P(green shirt) x P(blue pants) • If A and B are not independent, then • P(A and B) = P(A) x P(B | A) • For example, if I choose pants that look good with my shirt, then: P(green shirt and blue pants) • = P(green shirt) x P(blue pants, given the green shirt)

  13. A Numerical Example

  14. P(green shirt) • P(blue pants) • P(blue pants OR green shirt) • P(blue pants AND green shirt) • P(blue pants GIVEN green shirt) • P(blue pants GIVEN not-green shirt) • P(green shirt AND not-green shirt)

  15. P(green shirt) = 10/100 = .1 • P(blue pants) = 40/100 = .4 • P(blue pants AND green shirt) = 4/100 = .04 • P(blue pants OR green shirt) = .4+.1-.04 = .46 • P(blue pants GIVEN green shirt) = 4/10 = .4 • P(blue pants GIVEN not-green shirt) = 36/90 = .4 • P(green shirt AND not-green shirt) = zero

  16. A Numerical Example

  17. A Numerical Example

  18. A Numerical Example

  19. A Numerical Example (in which shirts and pants are not independent)

  20. P(green shirt) • P(blue pants) • P(blue pants AND green shirt) • P(blue pants OR green shirt) • P(blue pants GIVEN green shirt) • P(blue pants GIVEN not-green shirt) • P(green shirt AND not-green shirt)

  21. P(green shirt) = 10/100 = .1 • P(blue pants) = 40/100 = .4 • P(blue pants AND green shirt) = 8/100 = .08 • P(blue pants OR green shirt) = .4 + .1 - .08 = .42 • P(blue pants GIVEN green shirt) = 8/10 = .8 • P(blue pants GIVEN not-green shirt) = 32/90 = .356 • P(green shirt AND not-green shirt) = zero

  22. A Numerical Example (in which shirts and pants are not independent)

  23. THE ADDITION RULE for more than two disjoint events • If A and B and C are mutually disjoint, then • P(A or B or C) = P(A) + P(B) + P(C) • P(green or blue or white shirt) • = P(green shirt) + P(blue shirt) + P(white shirt)

  24. THE MULTIPLICATION RULE for more than two independent events • If A and B and C are mutually independent, then • P(A and B and C) = P(A) x P(B) x P(C) • If I pick shirts, pants, and belts independently: • P(green shirt and blue pants and black belt) • = P(green shirt) x P(blue pants) x P(black belt)

  25. Homework 5 • 4.1 (1 or 2 or 3), 8 • 4.2 11, 14, 20, 28 • 4.5 92, 96, 105

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