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.

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

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.

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)

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)

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.

Independence uEvents 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)

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.

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)

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)

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)