What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1)
Probability - Terminology random (probability) experiments a process that leads to well defined results called outcomes outcomes the result of a single trial of an experiment aka: elements, sample points eg experiment: outcome: Section 2 - Probability (2)
Probability - Terminology sample space the set of all possible outcomes in a random experiment a listing of all sample points eg - flip a coin twice event a subset of the sample space with some special characteristics eg - get a head first - get at least one tail Section 2 - Probability (3)
Probability - Terminology Venn diagram a graphical representation of probability problems S A B A = (HH, HT) B = (HT, TH, TT) Section 2 - Probability (4)
1. classical probability uses sample spaces to determine the numerical probability that an event will happen a sample space has n equally likely outcomes; an event contains f of those ie ( E) P = f n eg Section 2 - Probability (5)
2. empirical probability uses observations (frequency distributions) to determine numerical probabilities eg flip a coin 500 times... P(head) = frequency distributions... P(belonging to a class) = Section 2 - Probability (6)
3. subjective probability educated guess, opinions, inexact information based on a person s knowledge, experience, or evaluation of a situation eg Section 2 - Probability (7)
3. subjective probability odds if E is an event... - the odds in favour of E are: ( E) odds = P( E) P( E ) - the odds against E are: ( E ) odds = P( E) P( E) Section 2 - Probability (8)
Laws of Probability 1all outcomes in a sample space are equally likely to occur (ie, each has the same probability) P( E) = α where: α = 0 α 1 Section 2 - Probability (9)
Laws of Probability 2an event s complement is when that event does not occur S _ E E P ( E) + P( E ) = P( S ) = 1 1 ( ) ( ) P E = P E eg B = (HT, TH, TT) Section 2 - Probability (10)
Probability - Compound Events experiment: roll a die S = (1, 2, 3, 4, 5, 6) 3 events: roll an even number A = (2, 4, 6) roll less than 3 B = (1, 2) roll a 1 C = (1) S B A C Section 2 - Probability (11)
Probability - Compound Events S B A intersection of events: eg probability of events occurring simultaneously, (overlapping elements) P ( A B) = P ( A) P( B) multiplication rule for independent events Section 2 - Probability (12)
Probability - Compound Events C A S mutually exclusive or disjoint events: occurrence of one event precludes the other, no intersection (no common elements) Section 2 - Probability (13)
Probability - Compound Events C A S union of events: eg probability of one event or another (others) occurring ( A C) = P( A) P( C) P + addition rule for mutually exclusive events Section 2 - Probability (14)
Probability - Compound Events S B A union of events that are not mutually exclusive: eg A = (2, 4, 6) P(A) = 1/2 B = (1, 2) P(B) = 1/3 Section 2 - Probability (15)
Probability - Compound Events B A S union of events that are not mutually exclusive: eg P ( A B) = P( A) + P( B) P( A B) = P ( A) + P( B) P( A) P( B) addition rule for events that are not mutually exclusive Section 2 - Probability (16)
Probability - Compound Events Review: P (Queen of Hearts) = P (Queen of Hearts or Queen of Spades) = P (a Queen or a Heart) = Section 2 - Probability (17)
Laws of Probability (review) 1general formula ( E) P = f n = α where 2complement law α = 0 α 1 P ( E) + P( E ) = P( S) = 1 1 P = ( E) P( E ) Section 2 - Probability (18)
Laws of Probability (review) 3addition law (for mutually exclusive events) ( A B) = P( A) P( B) P + 4generalized addition law P (events need not be mutually exclusive) ( A B) = P( A) + P( B) P( A B) = P ( A) + P( B) P( A) P( B) Section 2 - Probability (19)
Laws of Probability 5multiplication law (for independent events) P ( A B) = P A ( ) ( ) P B independent events - the probability of one event occurring does not effect the probability of the other occurring. dependent events - the outcome of the second event is affected in such a way that the probability is changed. Section 2 - Probability (20)
Laws of Probability 6generalized multiplication law (events need not be independent) P ( A B) = P ( A) P( B A) conditional probability the probability that event B occurs after (given that) event A has already occurred. if events A and B are independent, then... P(B A) = P(B) Section 2 - Probability (21)
Probability Review: - drawing from a deck of cards without replacement P (Q, K) = P (Q, Q) = P (Q, Q, Q) = P (J, Q, K) = P (Heart, Diamond) = P (Heart, Heart, Heart) = Section 2 - Probability (22)
Probability log sort problem: A mill purchases logs from 2 suppliers: companies loggers 80% of logs 20% of logs 1% defective 2% defective If all of the logs are randomly put in a log sort, what is the probability that a randomly selected log will be defective? Section 2 - Probability (23)
Probability log sort problem (cont.): Section 2 - Probability (24)
Probability log sort problem (cont.): Section 2 - Probability (25)
Probability log sort problem (cont.): Section 2 - Probability (26)
Conditional Probability P ( A B) = P ( A) P( B A) P ( B A) = P ( A B) P( A) eg flip a coin twice S = (HH, HT, TH, TT) 2 events: at least one Head: A = (HH, HT, TH) first toss a Head: B = (HH, HT) What is P(B A)? Section 2 - Probability (27)
Conditional Probability log sort problem: 1,000 logs: company logger TOTAL defective 8 4 12 not defective 792 196 988 TOTAL 800 200 1000 What is the probability of selecting a defective log given that it came from a logger? Section 2 - Probability (28)
Conditional Probability log sort problem (cont.): Section 2 - Probability (29)
Conditional Probability has order, depending on the question being asked... 1natural order - ordered in terms of events ie eg P (B A) - B occurs after A P (D C) 2Bayesian probability - reverse order - based on some information, what is the probability of an event which occurred in the past? eg P (C D) Section 2 - Probability (30)
Conditional Probability 2Bayesian probability (cont.) - solve using conditional probability Section 2 - Probability (31)
Conditional Probability species blow-down problem: species: blow-down rate: D-fir 0.0001 cedar 0.05 hemlock 0.01 Given a tree that has blown down, what is the probability that it is a cedar? Section 2 - Probability (32)
Conditional Probability species blow-down problem (cont.): Section 2 - Probability (33)
Conditional Probability species blow-down problem (cont.): Section 2 - Probability (34)
Conditional Probability 2Bayesian probability (cont.) Bayes Theorem: for 2 events, A and B, where B follows A: P ( A B) j = n P j= 1 ( A B) P j ( A B) j = n P j= 1 ( A ) P( B A ) P j ( A ) P( B A ) j j j Section 2 - Probability (35)