Chapters 14/15 AP Statistics Mrs. Wolfe PROBABILITY RULES Chapters 14/15 AP Statistics Mrs. Wolfe

What is probability? Probability is the long run relative frequency of some occurrence Probability of event A = P(A) P(A)= number of successful outcomes number of total outcomes

Example 1: Imagine that a bowl contains the following Marbles: 4 green, 2 blue, 3 white, 1 red. P(Green) = ? P(Red or blue)= ? P(not white) = ? P (green and blue)= ?

Important words in Probability OR—means to add also means the UNION(U) of sets A and B P(A U B) = P(A) + P(B)- P(A∩ B)

AND--means to multiply 2nd Important Word AND--means to multiply also means the INTERSECTION (∩) of sets A and B P(A ∩ B) = P(A) ∙ P(B) assumes A and B are INDEPENDENT

NOT—means to subtract from 1 or from 100% 3rd Important Word NOT—means to subtract from 1 or from 100% A c or Ā or A’=the complement of A P(Ac) = 1 – P(A)

KEY VOCABULARY Trial—a single attempt of a random occurrence Outcome—the value measured or observed for an individual trial Event—a collection of outcomes-designated with capital letters A, B, C, etc. Sample Space—collection of all outcomes possible

TYPES OF PROBABILITY Theoretical—mathematical computation involved to determine what should happen Empirical—experiment performed to count chance of event happening Subjective—educated guess –no real probability theory used

LAW OF LARGE NUMBERS In the long run, empirical probability will settle down toward the theoretical probability. Does not mean that the “Law of Averages” will be true…just because a coin hasn’t been heads in 5 tries, it still has a 50% chance of being heads on the next try!

PROPERTIES OF PROBABILITY Probability of all outcomes must sum to 1. 2. 0 < P(x) < 1

EXAMPLE USING VENN DIAGRAMS Police reports that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test and 22% are given both tests. What is the probability of a. a test? b. a blood test or a breath test, but not both? c. neither test?

Important Event Concepts Disjoint Events—also called mutually exclusive means that A and B share no common outcomes Independent Events—how one event occurs does not affect the probability of the second event

Independent? Mutually exclusive? Given P(A) = 0.6 and P(B) = 0.3. P (A U B) = 0.75. Are A and B mutually exclusive? P(A ∩ B) = 0.18. Are A and B independent?

Blood type problem 45% type O, 40% type A, 11% type B, 4% type AB What is the probability of a person being Type A and Type O? What is the probability of two persons in a row being Type A?

Car Problem Review Suppose 40% of cars in your area are manufactured in the US, 30% in Japan, 10% in Germany and 20% in other countries. If a car is selected at random, find a. P(car not US made) b. P(car from Japan or Germany) c. P(two cars in a row from Japan) d. P(at least one of three cars is US made)

CONDITIONAL PROBABILITY-CHART Public Private Total Male 180 110 290 Female 95 55 150 Totals 275 165 440