Chapter 14 – From Randomness to Probability

Probability Around Us 20% chance of rain today 1 in 10 bottle caps wins a free soda (odds) Will you hit traffic today on the way home? Guessing on a multiple choice question Getting a flush in a poker hand Should you take out collision insurance? Patterns/coincidences Same artist comes up 2 songs in a row on mp3 player 2 people born on the same day

Types of probability Subjective Empirical Theoretical No way to measure, just a guess/estimate Empirical Observed probability Can use experiments or simple observations Theoretical Probabilities that can be calculated exactly

Terminology Trial: each occasion that a random phenomenon is observed Outcome: result of trial Event: combination of results Sample Space: all possible outcomes

Example Roll 2 dice, look at the sum and see if it’s odd Trial: each roll of 2 dice Outcome: sum of each roll of 2 dice Event: sum is odd Sample Space: all possible rolls of 2 dice (36 in total)

Law of Large Numbers (LLN) For independent trials, as the number of trials increases, the long-run relative frequency of repeated events gets closer and closer to a single value. Relative frequency observed is empirical probability

Nonexistent Law of Averages LLN only applies to long-term observations Outcomes are not “due” to happen to even things out Long-term observations happen over a very long time

Modeling Probability P(A) = Outcomes need to be equally likely! P(heads) P(roll a 6) P(face card) P(student at random is male)

Formal Probability All probabilities are between 0 and 1: 0 ≤ P(A) ≤ 1 Set of all possible outcomes has probability of 1 P(S) = 1 Complement of A: Ac P(A) = 1 – P(Ac)

Addition Rule (simple version) Assuming A and B are disjoint (mutually exclusive) events, P(A or B) = P(A) + P(B) P(2 or Q from a deck of cards) P(4 or 5 on a single die)

Multiplication Rule (simple version) For two independent events A and B, P(A and B) = P(A) x P(B) P(flip 3 coins, 3 Heads) P(draw 2 cards with replacement, 2 face cards) P(draw 2 cards with replacement, neither face cards) P(flip 3 coins, at least 1 Heads)