HUDM4122 Probability and Statistical Inference February 18, 2015.

HUDM4122Probability and Statistical Inference

February 18, 2015

HW

• Getting harder…

Difficulties

A lot of trouble with sample space calculation

A reminder

• The sample space is the total number of combination of things that can happen

• If you flip a fair coin twice, the sample space is 4: HHTH HT TT

Problem 1

• What is the sample space if you flip a coin 6 times?

• Correct answer: 2x2x2x2x2x2 = 64• Most common answer: 12• Also: 1/64

Problem 2

• You have made friends with a specially trained mouse, who, on a given step, randomly goes left 1/3 of the time, forwards 1/3 of the time, and right 1/3 of a time. If the mouse takes 7 steps, what is the sample space?

• Correct answer: 3x3x3x3x3x3x3• Common wrong answer: 21

Problem 5

• You and your friends order a pizza with 8 slices. One of the slices, for some obscure reason, has anchovies. You *HATE* anchovies. Before you get to the pizza, each of your 7 friends takes a single slice, apparently at random. What is the sample space of this meal?

• Correct answer: 8*7*6*5*4*3*2*1• Common wrong answers: 8, 8^7– Why are these wrong?

Probability Calculation

Problem 6

• You and your friends order a pizza with 8 slices. One of the slices, for some obscure reason, has anchovies. You *HATE* anchovies. Before you get to the pizza, each of your 7 friends takes a single slice, apparently at random. What is the probability that you end up with anchovies?

• Correct answer: 1/8• Common wrong answer: 1/40320

Problem 8

• You and your friends order a pizza with 8 slices. One of the slices, for some obscure reason, has anchovies. You *HATE* anchovies. Before you get to the pizza, 2 of your 7 friends take a single slice, apparently at random. They both did not get anchovies. What is the probability that you end up with anchovies?

• Correct answer: 0%– Why?

• Common wrong answers: 1/6, 1/5

Problem 11

• 21% of Americans went to an art gallery or museum in the last year. 23% of Americans went to a baseball game last year. 4% of Americans went to both (I totally made that last one up). What percent of americans went to a art gallery OR a museum OR a baseball game last year?

• What’s the answer?

Problem 11

• 21% of Americans went to an art gallery or museum in the last year. 23% of Americans went to a baseball game last year. 4% of Americans went to both (I totally made that last one up). What percent of americans went to a art gallery OR a museum OR a baseball game last year?

• Correct answer: 40%• Common wrong answers: 44%, 48%

Problem 14

• The probability that a New Yorker takes the subway is 37%. Let's say that the probability that a New Yorker goes to a museum or gallery each year is 34%. The probability that a New Yorker goes to a museum or gallery each year, if they take the subway, is 41%. What is the probability that a New Yorker takes the subway AND goes to a museum or gallery each year?

• Correct answer: 15%• Common wrong answer: 41%– Why is this wrong?

Problem 14

• The probability that a New Yorker takes the subway is 37%. Let's say that the probability that a New Yorker goes to a museum or gallery each year is 34%. The probability that a New Yorker goes to a museum or gallery each year, if they take the subway, is 41%. What is the probability that a New Yorker takes the subway AND goes to a museum or gallery each year?

• Correct answer: 15%• Another common wrong answer: 13%– Why is this wrong?

Problem 15

• The probability that a student passes "Intro to Basketweaving" is 72%. The probability that a student passes "Intro to Psychoceramics" is 21% if they fail "Intro to Basketweaving", and is 94% if they pass "Intro to Basketweaving". What is the probability that a student passes both classes?

• Why is the correct answer 68% rather than 20%?

Combinations and Permutations

Problem 10

• Professor Padeiro owns 7 computers. She wants to take 3 of them with her on a trip. How many combinations of computers could she take?

• What is the correct answer?

Problem 10


• What is the correct answer?=35

Problem 10


• What is the correct answer?=35

• Common wrong answer = 210

What we didn’t cover last time

General Multiplication Rule

• What if A and B are independent?

• Like two flips of a fair coin

General Multiplication Rule

• What if A and B are independent?

• Like two flips of a fair coin

• In that case, P(B|A)=P(B)

Multiplication Rule For Independent Events

• If A and B are independent

Multiplication Rule For Independent Events

• If A and B are independent

• This is the rule we were using, when we computed…– Multiple coin flips– Multiple rolls of a 6-sided die

Any last comments or questions for the day?

Today

• Ch. 4.7 in Mendenhall, Beaver, & Beaver

Today

• Bayes’ Rule

Today

• Bayes’ Rule

• Also (more frequently) called– Bayes’ Theorem– Bayes’ Law

Very Important Rule in Statistics

• Underpins Bayesian Statistics– One of the two core branches of Statistics– Not a focus of this class, which is focused on the

other branch, Frequentist statistics

• Underpins major areas of Data Mining and Machine Learning– Including core methods of educational data

mining, such as Bayesian Knowledge Tracing

Classic Version

𝑃 ( 𝐴|𝐵 )=𝑃 (𝐵|𝐴 )𝑃 (𝐴)

𝑃 (𝐵)

Let’s apply it

• P(B|A) = 0.4• P(A) = 0.7• P(B) = 0.3• P(A|B)=?

Let’s apply it

• P(B|A) = 0.4• P(A) = 0.7• P(B) = 0.3• P(A|B)= 0.93

Example

• Maria is using Reasoning Mind software to learn mathematics

• If she knows a skill, there’s a 60% chance she gets the problem right

• There’s a 40% chance she knows the skill• There’s a 70% chance she gets the problem right

• What’s the probability that if she gets the problem right, she knows the skill?

A = knows skill, B = gets problem right





A = knows skill, B = gets problem right


• If she knows a skill, there’s a 60% chance she gets the problem right. P(B|A)

• There’s a 40% chance she knows the skill. P(A)• There’s a 70% chance she gets the problem right. P(B)


Example




• What’s the probability that if she gets the problem right, she knows the skill?– 34.2%

Be careful…

• About what your A is• And what your B is

Do this one in pairs

• Dan is taking an online course using the Purdue Course Signals platform, that detects when a student is at-risk of failing the course

• If he is at-risk, there’s a 80% chance he skips the first homework• There’s a 50% chance he is at-risk• There’s a 60% chance he skips the first homework

• What’s the probability that if he skips the first homework, he is at-risk?


• Dan is taking an online course using the Purdue Course Signals platform, that detects when a student is at-risk of failing the course

• If he is at-risk, there’s a 80% chance he skips the first homework

• There’s a 50% chance he is at-risk• There’s a 60% chance he skips the first homework

• What’s the probability that if he skips the first homework, he is at-risk?– 66.7%


• The Yonkers College of Holistic Phrenology just had an unspeakably embarrassing scandal

• Historically, among colleges of this type that are denied accreditation, 20% have had a recent scandal

• There’s a 4% chance of a college of this type being denied accreditation

• There’s a 1% chance of a college of this type having a scandal

• Given that this college just had a scandal, what is the probability it will be denied accreditation?


• The Yonkers College of Holistic Phrenology just had an unspeakably embarrassing scandal

• Historically, among colleges of this type that are denied accreditation, 20% have had a recent scandal

• There’s a 4% chance of a college of this type being denied accreditation

• There’s a 1% chance of a college of this type having a scandal

• Given that this college just had a scandal, what is the probability it will be denied accreditation?– 80%

Questions? Comments?

Where did Bayes’ Rule come from?


P(Actually Bayes) = 0.1


• Simple to derive

Recall General Multiplication Rule

Note Also That

Which Means That

• P(B)P(A|B) =

Divide Both Sides by P(B)

This was the Classic Version of Bayes’ Rule

• When people talk about Bayes’ Rule, they generally mean this version

This was the Classic Version of Bayes’ Rule

• When people talk about Bayes’ Rule, they generally mean this version

• There is also a General Version seen in the book

Before we get there…

• Law of Total Probability

Law of Total Probability

Example

• I’m wondering whether a student will quit the problem set before completing

• The student might be (exhaustively, mutually exclusively)– Working in System Appropriately– Gaming the System– Getting Answers From a Friend

Example


• The student might be (exhaustively, mutually exclusively)– P(Working in System Appropriately) = 0.7– P(Quit | WSA) = 0.02

– P(Gaming the System) = 0.1– P(Quit | GS) = 0.01

– P(Getting Answers From a Friend) = 0.2– P(Quit | GAFF) = 0.3

Note that P(WSA)+P(GS)+P(GAFF) = 1





Note that P(WSA)+P(GS)+P(GAFF) = 1If this isn’t true,

it’s not exhaustive or not mutually exclusive





Example

• P(Quit) = P(Working in System Appropriately) * P(Quit | WSA) +P(Gaming the System) * P(Quit | GS) +P(Getting Answers From a Friend) * P(Quit | GAFF)

Example


• P(Quit) = 0.7*0.02+ 0.1*0.01 + 0.2*0.3

Example


• P(Quit) = 0.7*0.02+ 0.1*0.01 + 0.2*0.3 = 0.014+0.001+0.06 = 0.075


• I’m wondering whether my kiddo has stomach flu

• The kiddo might be (exhaustively, mutually exclusively)– P(Puking) = 0.05– P(Stomach Flu | Puking) = 0.4

– P(Not Puking) = 0.95– P(Stomach Flu | Not Puking) = 0.01

• Your answer?

If we take the law of total probability

• And compare it to the multiplied probability coming out of one event…

• We get…

Extended Form of Bayes’ Rule

• What the book simply refers to as Bayes’ Rule(but most people don’t)

Extended Form of Bayes’ Rule

• ) =

• A is an event• S1 through Sk represent a group of mutually

exclusive and exhaustive sub-populations

Example

• In ASSISTments, on the first attempt at problem P, a student can request a hint, give a common incorrect answer, give an uncommon incorrect answer, or give a correct answer

• P(hint) = 0.3• P(common incorrect) = 0.2• P(uncommon incorrect) = 0.4• P(correct) = 0.1

Example


• P(knows skill | hint) = 0.1• P(knows skill | common incorrect) = 0.2• P(knows skill | uncommon incorrect) = 0.1• P(knows skill | correct) = 0.7




• ) =




• ) = =0.39

Comments? Questions?

Any last comments or questions for the day?

Upcoming Classes

• 2/23 Discrete Random Variables and Their Probability Distributions– Ch. 4-8

• 2/25 Binomial Probability Distribution– Ch. 5-2– HW 4 due

Homework 4

• Due in 7 days• In the ASSISTments system

HUDM4122 Probability and Statistical Inference February 18, 2015.

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