Probability and Statistical Inference Practice Problems
Explore various probability practice problems including sample space calculation, probability calculations, and scenarios involving random events like flipping coins and sharing pizza slices with friends. Test your understanding of sample space theory and probability concepts through different scenarios and calculations.
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Presentation Transcript
HUDM4122 Probability and Statistical Inference February 18, 2015
HW Getting harder
HW Getting harder
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: HH TH 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?
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%?
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 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? 7! 3!4!=35
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? 7! 3!4!=35 Common wrong answer = 210
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 ? ? ? ?(?) ?(?) ? ? ? =
Lets apply it ? ? ? ?(?) ?(?) ? ? ? = P(B|A) = 0.4 P(A) = 0.7 P(B) = 0.3 P(A|B)=?
Lets 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 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 Maria is using Reasoning Mind software to learn mathematics 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) What s the probability that if she gets the problem right, she knows the skill?
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? 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?
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? 66.7%
Do this one in pairs 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?
Do this one in pairs 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%
Where did Bayes Rule come from? P(Actually Bayes) = 0.1
Where did Bayes Rule come from? Simple to derive
Recall General Multiplication Rule ? ? ? = ? ? ?(?|?)
