Lecture PowerPoint Slides Basic Practice of Statistics 7 th Edition

- NIH Ethics Certification (required; due next Friday) - Next Spreadsheet Assignment due March 16 - Chapter 12 Moodle Quiz (optional) - Questions on Moodle Quiz 9? - Let’s take a quick look at Spreadsheet Assignment 7

The idea of probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. RANDOMNESS AND PROBABILITY We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Big probability question in statistics Is this event likely to have happened by chance (randomness), or is something else going on?

Tea firstMilk first Smart lady?

The test: 8 cups of tea “Null” hypothesis: The lady does not have the ability to tell. But getting all 8 correct just by guessing is pretty unlikely, so if she can do it, we might be willing to reject our null hypothesis and believe her.

Can you pass the “lady tasting tea” test?

Find your partner and grab a white board

Estimate the probability of… The top card of a well-shuffled deck of cards being the ace of hearts You knowing the first name of a randomly selected student at EMU For a randomly chosen person, the probability that they will find someone who they love and who loves them You knowing about the topic of a randomly chosen Wikipedia article A randomly selected Black male spending time in prison at some point in his life (Bureau of Justice Statistics, bjs.gov) The British lady passing the tea tasting test by chance

Two probabilities that we will often be working with are 0.95 and What are some real life events that have (very roughly) these probabilities?

When rolling a 6-sided dice one time, what’s the chance of getting either a 2 or a 5? (1/6) + (1/6) = 2/6

When rolling a 6-sided dice one time, what’s the chance of getting either an even number or a number less than 4? (3/6) + (3/6) = 6/6? NO!

Sum Law of Probability The probability of at least one of two events happening is the probability of each one happening minus the probability of both of them happening. P(A or B) = P(A) + P(B) – P(A and B)

When rolling a 6-sided dice one time, what’s the chance of getting either an even number or a number less than 4? (3/6) + (3/6) – (1/6) = 5/6

When rolling two 6-sided dice, what’s the chance of rolling an 11?

Probability models Sample Space 36 Outcomes Sample Space 36 Outcomes Since the dice are fair, each outcome is equally likely. Each outcome has probability 1/36. Since the dice are fair, each outcome is equally likely. Each outcome has probability 1/36. Example: Give a probability model for the chance process of rolling two fair, six-sided dice―one that’s red and one that’s green.

Benford’s Law Let’s say you had a dataset with the populations of every village, town, and city in the U.S. Now, for each population, keep the first digit and throw the rest away. What portion of these first digits would be 7s? 1s? 9s?

Something like this?

Benford’s law

Benford’s law

Benford’s law

The moral: When calculating a probability, don’t automatically assume that all outcomes are equally likely!

When rolling a 6-sided dice three times, what’s the chance of getting an even number all three times? (1/2)*(1/2)*(1/2) = 1/8

Multiplication Law of Probability If A and B are independent events, then the probability that both will happen is the product of the individual probabilities. P(A and B) = P(A)P(B)

Gambler’s fallacy

Unnumbered art Page 298, The Basic Practice of Statistics, © 2015 W. H. Freeman, Darrell Walker/HWMS/Icon SMI/Newscom Hothand fallacy

When rolling a 6-sided dice two times, what’s the chance of getting doubles? 6*(1/6)*(1/6) = 6/36 = 1/6 (1)*(1/6) = 1/6

When rolling a 6-sided dice two times, what’s the chance of getting at least one 1? 1 – (5/6)*(5/6) = 11/36

Complement Law The probability of an event not happening is 1 minus the probability of the event happening. (Example: If there’s a 30% chance of rain tomorrow, then there’s a 70% chance there won’t be rain tomorrow, because 1 – 0.30 = 0.70)

Probability rules 1. Any probability is a number between 0 and All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. 4. The probability that an event does not occur is 1 minus the probability that the event does occur. The probability that an event occurs and the probability that it does not occur always add to 100%, or 1.

Factorials : n! = n*(n-1)*(n-2) … (2)*(1) Example: 5! = 5 * 4 * 3 * 2 * 1

When rolling a 6-sided dice two times, what’s the chance of getting double 1s? (1/6) * (1/6) = 1/36

When rolling a 6-sided dice three times in a row, what’s the chance of getting a run? (for example, a 3, then a 4, then a 5) – (5,4,3 is not a run) (4/6)*(1/6)*(1/6) = 4/216

When rolling a 6-sided dice three times in a row, what’s the chance of getting three different numbers? (i.e., you don’t roll the same number twice) (1)*(5/6)*(4/6) = 20/36

Counting: The hardest part of a probability is often figuring out the number of ways to get a “success”. This is called counting.

How many arrangements are there for the letters in MATH? 4*3*2*1 = 4! = 24

How many arrangements are there for the letters in GOSPEL? 6! = 720

How many arrangements are there for the letters in MOMMY? 5! / 3! = 20

How many arrangements are there for the letters in STATISTICS? 10! / (3!3!2!) = 50,400

How many 3-letter arrangements are there for the letters in GOSPEL? 6*5*4 = 120

How many 2-letter arrangements are there for the letters in STATISTICS? 10*9 - (2!2!2!) = 82 ways

How many 2-letter arrangements are there for the letters in STATISTICS? 10*9 - (2!2!2!) = 82 ways WRONG!!

How many 2-letter arrangements are there for the letters in STATISTICS? Case 1: C or A first = 2*4 = 8 ways Case 2: I,T,S first = 3*5 = 15 ways Total = = 23 ways

How many ways can 6 people sit around a circular table? 1 person sits down anywhere; 5 choices for the spot to the left, etc. 5*4*3*2*1 = 5! = 120 ways

How many ways can 3 guys and 3 girls sit around a circular table if they must alternate? 1 guy sits down anywhere; 3 choices for the spot to the left, etc. 3*2*2*1*1 = 12 ways

How many ways can 6 people sit around a circular table? 1 person sits down anywhere; 5 choices for the spot to the left, etc. 5*4*3*2*1 = 5! = 120 ways

How many ways can 6 people sit around a circular table if there are 2 people who can NOT sit by each other? 5! – (2*(4!)) = 72 ways

How many natural numbers less than 10,000 have no 5s in them? (9^4) - 1 = 6,560 numbers

Search for randomness* Computer programs, including applets Random number table Physical means, such as tossing coins or rolling dice “Chaos theory” Quantum mechanics

Fill in the 20 boxes as randomly and as quickly as possible HTHHTHTTTH THTHHHTHTT

Now repeat the experiment with actual flips of a coin