Understanding Randomness and Probability: Key Concepts Explained
This text explores the foundational principles of randomness and probability. It clarifies the distinction between random events and haphazard occurrences, explaining how randomness behaves predictably over a large number of trials. Key definitions such as sample space, events, probabilities, and rules like the Addition Rule and Multiplication Rule are detailed. The text also covers chances and odds, providing examples to illustrate these concepts, and explains the importance of independence in probabilities, making it an essential guide for anyone looking to grasp these fundamental statistical ideas.
Understanding Randomness and Probability: Key Concepts Explained
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Presentation Transcript
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 • 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)
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)
