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RANDOMNESS

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.

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RANDOMNESS

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  1. 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.

  2. 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

  3. 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.

  4. 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)

  5. 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)

  6. 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.

  7. Independence • Events A and B are independent if the probability of event B is not affected by A’s occurring or not occurring: • 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 after H1) = P(H2 after T1) = P(H2)

  8. 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.

  9. 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)

  10. 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)

  11. 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)

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