Stat 155, Section 2, Last Time Producing Data: How to Sample? –Placebos –Double Blind Experiment –Random Sampling Statistical Inference –Population “parameters”,, –Sample “statistics”,, (keep these separate) Probability Theory

Reading In Textbook Approximate Reading for Today’s Material: Pages , Approximate Reading for Next Class: Pages ,

Chapter 4: Probability Goal: quantify (get numerical) uncertainty Key to answering questions above (e.g. what is “natural variation” in a random sample?) (e.g. which effects are “significant”) Idea: Represent “how likely” something is by a number

Probability Recall Basics: Assign numbers (representing “how likely”), to outcomes E.g. Die Rolling: P{comes up 4} = 1/6 Outcome is “4” Probability is 1/6

Simple Probability Quantify “how likely” outcomes are by assigning “probabilities” I.e. a number between 0 and 1, to each outcome, reflecting “how likely”: Intuition: 0 means “can’t happen” ½ means “happens half the time” 1 means “must happen”

Simple Probability Main Rule: Sum of all probabilities (i.e. over all outcomes) is 1: E.g. for die rolling:

Simple Probability HW: 4.13a 4.15

Probability General Rules for assigning probabilities: i.Frequentist View (what happens in many repititions?) ii.Equally Likely: for n outcomes P{one outcome} = 1/n (e.g. die rolling) iii.Based on Observed Frequencies e.g. life tables summarize when people die Gives “prob of dying” at a given age “life expectancy”

Probability General Rules for assigning probabilities: iv.Personal Choice: –Reflecting “your assessment” –E.g. Oddsmakers –Careful: requires some care (key is prob’s need to sum to 1) HW: 4.19

Probability - Events More Terminology (to carry this further): An event is a set of outcomes Die Rolling: “an even #”, is the event {2, 4, 6} Notes: –If betting on even don’t care about #, only even or odd –Thus events are our foundation –Each outcome is an event: the set containing just that outcome –So event is the more general concept

Probability on Events Sample Space is the set of all outcomes = = “event with everything that can happen” Extend Probability to Events by: P{event} = sum of probs of outcomes in event

Probability Technical Summary: A probability model is a sample space I.e. set of outcomes, plus a probability, P P assigns numbers to events, Events are sets of outcomes

Probability Function The probability, P, is a “function”, defined on a set of events Recall function in math: plug-in get out Probability: P{event} = “how likely”

Probability Function E.g. Die Rolling Sample Space = {1, 2, 3, 4, 5, 6} “an even #” is the event {2, 4, 6} (a “set”) P{“even”} = P{2, 4, 6} = = P{2} + P{4} + P{6} = = 1/6 + 1/6 + 1/6 = 3/6 = ½ Fits, since expect “even # half the time”

Probability HW HW: b 4.17

And now for something completely different Did you here about the constipated mathematician?

And now for something completely different Did you here about the constipated mathematician? He worked it out with a pencil!

And now for something completely different Did you here about the constipated mathematician? He worked it out with a pencil! Apologies for the juvenile nature…

And now for something completely different Did you here about the constipated mathematician? He worked it out with a pencil! Apologies for the juvenile nature… But there is an important point:

And now for something completely different Did you here about the constipated mathematician? He worked it out with a pencil! Apologies for the juvenile nature… But there is an important point: The pencil is a powerful mathematical tool

And now for something completely different The pencil is a powerful mathematical tool An old student: –I was once “good in math” –But suddenly lost that –Reason: tried to do too much in head –Reason: never learned power of the pencil

And now for something completely different The pencil is a powerful mathematical tool For us: now is time to start using pencil I do PowerPoint in class You use pencil on HW (and exams) Change in mindset, from Excel…

Probability Now stretch ideas with more interesting e.g. E.g. Political Polls, Simple Random Sampling 2 views: 1.Each individual equally likely to be in sample 2.Each possible sample is equally likely Allows for simple Probability Modelling

Simple Random Sampling Sample Space is set of all possible samples An Event is a set of some samples E.g. For population A, B, C, D –Each is a voter –Only 4, so easy to work out

S. R. S. Example For population A, B, C, D, Draw a S. R. S. of size 2 Sample Space = {(A,B), (A,C), (A,D), (B,C), (B,D), (C,D)} outcomes, i.e. possible samples of size 2

S. R. S. Example Now assign P, using “equally likely” rule: P{A,B} = P{A,C} = … = P{C,D} = = 1/(#samples) = 1/6 An interesting event is: “C in sample” = {(A,C),(B,C),(D,C)} (set of samples with C in them)

S. R. S. Example P{C in sample} = i.e. happens “half the time”.

S. R. S. Probability HW HW C10: Abby, Bob, Mei-Ling, Sally and Roberto work for a firm. Two will be chosen at random to attend an overseas meeting. The choice will be made by drawing names from a hat (this is an S. R. S. of 2). a.Write down all possible choices of 2 of the 5 names. This is the sample space. b.Random choice makes all choices equally likely. What is the probability of each choice? (1/10) c.What is the prob. that Sally is chosen? (4/10) d.What is the prob. that neither Bob, nor Roberto is chosen? (3/10)

Political Polls Example What is your chance of being in a poll of 1000, from S.R.S. out of 200,000,000? (crude estimate of # of U. S. voters) Recall each sample is equally likely so: Problem: this is really big (5,733 digits, too big for easy handling….)

Political Polls Example More careful calculation: Makes sense, since you are “equally likely to be in samples”

And now for something completely different. An interesting phone conversation…. Sound File

Probability Now have prob. models But still hard to work with E.g. prob’s we care about, such as “accuracy estimators”, need better tools Need to look more deeply

3 Big Rules of Probability Main idea: calculate “complicated prob’s” By decomposing events in terms of simple events Then calculating probs of these And then using simple rules of probabilty to combine

3 Big Rules of Probability Rule I: the not rule: P{not A} = 1 – P{A} Why? E.g. equally likely sample points: And more generally:

The “Not” Rule of Probability Text Book Terminology (sec. 4.2): not A = for “complement” (set theoretic term) (I prefer “not”, since more intuitive)

The “Not” Rule of Probability HW: Rework, using the “not” rule: 4.17b

3 Big Rules of Probability Rule II: the or rule: P{A or B} = P{A} + P{B} – P{A and B} Why? E.g. equally likely sample points: Helpful Pic:

Big Rules of Probability E.g. Roll a die, Let A = “4 or less” = {1, 2, 3, 4} Let B = “Odd” = {1, 3, 5} Check how rules work by calculating 2 ways: Direct: P{not A} = P{5, 6} = 2/6 = 1/3 By Rule I: P{not A} = 1 – P{A} = 1 – 4/6 = 1/3

The “Or” Rule of Probability A = “4 or less” = {1, 2, 3, 4} B = “Odd” = {1, 3, 5} Check how rule works by calculating 2 ways: Direct: P{A or B} = P{1, 2, 3, 4, 5} = 5/6 By Rule II: P{A or B} = = P{A} + P{B} – P{A and B} = = 4/6 + 3/6 – 2/6 = 5/6 (check!)

The “Or” Rule of Probability Seems too easy? Don’t really need rules for these simple things But they are the key to bigger problems Such as Simple Random Sampling HW: 4.86 (0.317)

The “Or” Rule of Probability E.g: A college has 60% Women and 40% smokers, and 50% women who don’t smoke. What is the chance that a randomly selected student is either a women or a non- smoker? (seems “>60%”, but twice? Must be < 100%, i.e. must be some overlap…)

College Women – Smokers E.g. P{W or NS} = P{W} + P{NS} = P{W & NS} (choice of letters make easy to work with) = (1 – 0.4) – 0.5 = 0.7 (answer is 70% Women or Non-Smokers) Note: rules are powerful when used together

More “Or” Rule HW HW: C11 A building company bids on two large projects. The president believes the chance of winning the 1 st is 0.6, the chance of winning the 2 nd is 0.5, and the chance of winning both is 0.3. What is the chance of winning at least one of the jobs? (0.8)

The “Or” Rule of Probability E.g. Events A & B are “mutually exclusive”, i.e. “disjoint”, when P{A & B} = 0 (i.e. no chance of seeing both at same time) Useful Pic: Then: P{A or B} = P{A} + P{B} Text suggests “new rule”, I say “special case”

The “Exclusive Or” Rule HW: C12 Choose an acre of land in Canada at random. The probability is 0.35 that it is forest, and 0.03 that it is pasture. a.What is the probability that the acre chosen is not forested? (0.65) b.What is the probability that it is either forest or pasture? (0.38) c.What is the probability that a randomly chosen acre in Canada is neither forest nor pasture? (0.62)