Probability

1. Probability Two-way tables, discrete & continuous random variables; OLI CIS Module 15 to Module 17

2. Probability... • Probability calculations are the basis for inference (making decisions about a population based on a sample). • What we learn inabout probabilitywill help us describe statistics from random samples & randomized comparative experiments later in the course.

3. 1-in-6 game… As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, “Please try again!” while others said, “You’re a winner!” The company advertised the promotion with the slogan “1 in 6 wins a prize.” The prize is a free 20-ounce bottle of soda, which comes out of the store owner’s profits. Seven friends each buy one 20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. The store owner is concerned about losing money from giving away too many free sodas. She wonders if this group of friends is just lucky or if the company’s 1-in-6 claim is inaccurate. In this Activity, you and your classmates will perform a simulation to help answer this question.

4. 1-in-6 game… For now, let’s assume that the company is telling the truth, and that every 20-ounce bottle of soda it fills has a 1-in-6 chance of getting a cap that says, “You’re a winner!” We can model the status of an individual bottle with a six-sided die: let 1 through 5 represent “Please try again!” and 6 represent “You’re a winner!”

5. 1-in-6 game… 1. Roll your die seven times to imitate the process of the seven friends buying their sodas. How many of them won a prize? Repeat 3 times. 2. Plot your three results on the board to create a class dot plot displaying the number of prize winners we got in Step 1 on the graph. 3.What percent of the time did the friends come away with three or more prizes, just by chance? Does it seem plausible that the company is telling the truth or did the friends just get lucky? Explain.

6. Whose book is this? Suppose that four friends (including Ariana Grande) get together to study at a doughnut shop for their next test in high school statistics. When they leave their table to go get a doughnut, the doughnut shop owner decides to mess with them (you know… because of Ariana’sdoughnut scandal) and makes a tower using their textbooks. Unfortunately, none of the students wrote their name in their book, so when they leave the doughnut shop, each student takes one of the books at random. When the students return the books at the end of the year and the clerk scans their barcodes, the students are surprised to learn that none of the four had their own book. How likely is it that none of the four students ended up with the correct book? … simulation time! 

7. On four equally-sized slips of paper, write “Student 1,” “Student 2,” “Student 3,” and “Student 4.” Likewise, on four equally-sized slips of paper, write “Book 1,” “Book 2,” “Book 3,” and “Book 4.” Place the four papers with the student numbers on your desk. Then shuffle the papers with book numbers and randomly place one paper on each ‘student.” If the book number matches the student number, this represents a student choosing his own book from the tower of textbooks. Count the number of students who get the correct book. Repeat this process three times. Then plot your results on the board to create a class dot plot. How likely is it for none of the students to end up with their own book? What if we were to do this entire simulation again. Would you expect to get the same exact results? Why or why not?

8. Investigating Randomness… & More Simulation • Pretend that you are flipping a fair coin. Without actually flipping a coin, imagine the first toss. Write down the result you see in your mind, heads (H) or tails (T), below. • Imagine a second coin flip. Write down the result below. • Keep doing this until you have recorded the results of 25 imaginary flips. Write all 25 of your results in groups of 5 to make them easier to read, like this: HTHTH TTHHT, etc.

9. Investigating Randomness… & More Simulation… • A run is a repetition of the same result. In the previous example, there is a run of two tails followed by a run of two heads in the first 10 coin flips. Read through your 25 imagined coin flips that you wrote above and find the longest run (doesn’t matter if it was heads or tails; just your longest run). • On the board, plot the length of the longest run you you got (within your 25 values) to create a class dot plot.

10. Investigating Randomness… & More Simulation • Now, use a random digits table, technology, or a coin to generate a similar list of 25 coin flips. Find the longest run that you have. • Now lets create another dot plot with this new data from the class. Plot the length of the longest run you got above.

11. Randomness… • The idea of probability is that randomness is predictable in the long run. Unfortunately, our intuition about randomness tries to tell us that random phenomena should also be predictable in the short run. • Probability Applet (www.whfreeman.com/tps5e)

12. Random Phenomenon... We call an event ‘random’ if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

13. Big Idea Chance behavior (random phenomenon) is unpredictable in the short run, but has a regular and predictable pattern in the long run. Individual outcomes are uncertain; but a regular distribution of outcomes emerges in a large number of repetitions. Probability of any outcome of random phenomenon is the proportion of times an outcome would occur in a very long series of repetitions. Probability is a long-term relative frequency (simulations very helpful).

14. Probability vs. Odds • Probability = • Odds =

15. Careful... • It makes no sense to discuss the probability of an event that has already occurred. • Meaningless to ask what the probability is of an already-flipped coin being a tail. It’s already been decided. • Probability: future event • Statistics: past event

16. Definition: Simulation is... • the imitation of chance behavior, based on a model that accurately reflects the phenomenon under consideration. • Examples include...

17. Simulation... • Why would we want to simulate a situation (rather than carry the event out in reality)? • Discuss with a partner for one minute.

18. Simulation… model mustmatch situation... • What model could we use to simulate the probability of a soon-to-be new-born baby being a girl or a boy? What couldn’t be used as a model to simulate this situation? Discuss for one minute.

19. Simulation... • ... can be an effective tool/method for finding the likelihood of complex results IF you have a trustworthy model. • If not (if model does not correctly describe the random phenomenon), probabilities derived from model will also be incorrect/worthless.

20. Probability Rules ... • All probabilities are values between 0 & 1 Consider event A: • Sum of probabilities of all outcomes = 1 S sample space P(S) = 1

21. Momentary detour... Examples of disjoint/mutually exclusive events include: • miss a bus; catch a bus • play chess; sleep • turn left; turn right • sit down; stand up Non-examples of disjoint/mutually exclusive events include: • listen to music; do homework • sleep; dream

22. Mutually Exclusive/Disjoint events are... • Events that cannot happen simultaneously • Other examples of mutually exclusive/disjoint events?

23. Another brief detour… “union” * The union of any collection of events is the event that at least one of the collection occur. * Symbol “U” * P(A or B or C) = P(A U B U C)

24. Back to the Probability Rules ... • If 2 events have no outcomes in common (disjoint/mutually exclusive) then the probability of one or the other occurring is the sum of their individual probabilities. P(A or B) = P(A) + P(B) (Addition Rule for Disjoint Events) Example: P (rolling a 2 or rolling an odd) Non-example: P (rolling a 4 or rolling an even)

25. …more examples P (A or B or C) = P(A U B U C) = P (A) + P (B) + P (C) only if events are disjoint A: freshman P(A) = 0.30 B: sophomore P(B) = 0.35 C: junior P(C) = 0.20 D: senior P(D) = 0.15 All disjoint events. P(B U C) = P(A U D) = P (A U B U C U D) =

26. Probability Rules ... • Probability that an event does not occur is one minus the probability that the event will occur (complement rule) P(Ac)= 1 - P(A) Example: P (person has brown hair) = 0.53 So, P (person does not have brown hair) = 1 – 0.53 = 0.47 What would = ? What would = ?

27. Probability Rules .... ... one more probability rule later... Stay tuned ...

28. Probability Rules Practice Distance learning courses are rapidly gaining popularity among college students. The probability of any age group is just the proportion of all distance learners in that age group. Here is the probability model: Are rules 1 & 2 satisfied above? Are the above groups mutually exclusive events? Why or why not? P ( 18-23 yr or 30-39 yr) = P (not being in 18-23 yr category) = P (24-29 yr& 30-39 yr) =

29. Caution... Be careful to apply the addition rule only to disjoint/mutually exclusive events P (queen or heart) = 4/52 + 13/52 (??) .... not disjoint... this probability rule would not be correct in this case; more on this in a minute...

30. Review/Preview... Mutually Exclusive/Disjoint • sleeping; playing chess • walking; riding a bike Overlapping Events (not mutually exclusive) • roll an even; roll a prime • select 12th grader; select athlete • choose hard-cover book; choose fiction

31. What if ... • What if events are not disjoint/mutually exclusive? i.e., they can occur simultaneously (overlapping events) • How do we calculate P(A or B)?

32. Data Collection Time…Which do you use?

33. More Two-Way Table Practice.. • Go to my website, Math 140 data, copy & paste hair color & need glasses/contacts into StatCrunch • Create a two-way table in StatCrunch • Stat, Tables, Contingency, With Data

34. General Addition Rule (disjoint or overlapping) P (A or B) = P (A) + P (B) – P (A and B) P (A U B) = P (A) + P (B) – P (A∩ B)

35. Pierced ears, anyone? Find the probability that a given student: • has pierced ears • is a male • is male and has pierced ears • is male or has pierced ears

36. Morale of the story? Be careful to apply the addition rule for mutually exclusive events only to disjoint/mutually exclusive events P (queen or heart) = 4/52 + 13/52 .... not disjoint... counted queen of hearts twice P (queen or heart) = 4/52 + 13/52 – 1/52 (think of a Venn diagram; overlap)

37. Venn Diagrams… • Event A and (b) A, B mutually exclusive/disjoint

38. Venn Diagrams • Intersection of A & B (and) • Union of A, B (or)

39. Venn Diagrams with our Facebook & Twitter Class Data...

40. Conditional Probability... Remember... Probability assigned to an event can change if we know that some other event has occurred (“given”)

41. Conditional Probability... P (A | B) is read “the probability of A given B” P (female) = versus P (female | 15-17 years) =

42. Conditional Probability... caution P (male | 18-24 yr) = P (18-24 yr | male) =

43. Formula… To find the conditional probability P (A | B) The conditional probability P (B | A) is given by

44. General Multiplication Rule for Any Two Events The joint probability that events A and B both happen is P (A ∩ B) = P (A) P (B|A) P (female and 15-17yr) =

45. 89/16,639 P(A ∩ B) = P(A) P(B|A) A = female B = 15-17 years = (9,321/16,639) x (89/9,321) = 89/16,639 ✓

46. Facebook & Twitter... Conditional probabilities... • Table • Venn Diagram

47. Tree diagram… About 27% of adult Internet users are 18 to 29 years old, another 45% are 30 to 49 years old, and the remaining 28% are 50 and over. The Pew Internet and American Life Project finds that 70% of Internet users aged 18 to 29 have visited a video-sharing site, along with 51% of those aged 30 to 49 and 26% of those 50 or older. 

49. Let’s try a tree diagram now...

50. Review/Preview .... Two events A & B are independent if knowing that one occurs does not change the probability that the other occurs. Examples: - Roll a die twice. What I roll the first time does not change the probability of what I will roll the second time. - Win at chess; win the lottery - Student on debate team; student on swim team So, if events A and B are independent, then P (A|B) = P(A) and likewise P (B|A) = P (B).