1. STAT 311: Introductory Probability http://mrburkemath.blogspot.com/2010/12/snow-days.html 1

2. Part 1: Randomness http://mikeess-trip.blogspot.com/2011/06/gambling.html 2

3. Applications of Probability Games of Chance Diagnosis of Diseases Engineering Biology/agriculture Physics/Chemistry Business Management Computer Science 3

4. Chapter 1: Outcomes, Events, and Sample Spaces http://www.thescientificcartoonist.com/?p=102 men 4

5. Sample Spaces: Examples For each of the following examples, determine the sample set and state the possible values for the elements of the sample set. 1.Tossing Coins: We toss a coin 3 times 2.Rolling two 4-sided dies 3.Lifetime of a light bulb 4.Genetics: Dominant (A=black hair) or recessive (a = red hair) 5

6. Events: Examples 1.Tossing Coins: 3 times a)Determine the event that there is only one Head. b)Identify in words the event: {HHH,TTT} 2.Rolling two 4-sided dies a)Determine the event that the sum of the two dice is 9. b)Determine the event that the difference between the numbers of the white and red dies is 1. c)Identify in words the event: {(x, x+1): x  {1,2,3} } 3.Lifetime of a light bulb a)Determine the event that the light bulb lasts between 100 and 110 hours. b)Identify in words the event: {x | x < 200} 4.Genetics: Dominant (A=black hair) or recessive (a = red hair) a)Determine the event that the hair color is black. b)Identify in words the event: {aa, aA, Aa} 6

7. Example 1.17: Pick ten songs A student hears 10 songs (in a random shuffle mode) on her music player, noting how many of these songs belong to her favorite type of music. a)What is the sample space? b)How many different events are there? c)What is the event, A, that none of the first three songs are her favorite type of music? How many outcomes are there? d)What is the event, B, that the even-numbered songs are from her favorite type of music? How many outcomes are there? e)What is the event of A ∩ B? How many outcomes are there? f)What is the event, C, that the last 5 songs are from her favorite type of music? How many outcomes are there? g)What is the event B ∩ C? How many outcomes are there? 7

8. DeMorgan’s Laws Theorem 1.22 DeMorgan’s first law For a finite or infinite collection of events, A 1, A 2, … Theorem 1.23 DeMorgan’s second law For a finite or infinite collection of events, A 1, A 2, … 8

9. Set Theory: Other Laws Distributive Laws Let A, B, and C be subsets of S. Then a) A ∩ (B U C) = (A ∩ B) U (A ∩ C) b) A U (B ∩ C) = (A U B) ∩ (A U C) Associative and Commutative Laws Let A, B, and C be subsets of S, Then a) A ∩ B = B ∩ A b) A U B = B U A c) A ∩ (B ∩ C) = (A ∩ B) ∩ C d) A U (B U C) = (A U B) U C 9

10. Example: Subsets, etc. Write down the set of events for each of the following: Rolling 2 4-sided dice: A: the red die is a 4B: the white die is a 2 C: The sum of the two dice is 4 D: the red die is a 3 a)A C f) A c \C b)A ∩ Cg) D U (B ∩ C) c)A U Ch) (D U B) ∩ (D U C) d)A C ∩ C e)A C U C 10

11. Chapter 2: Probability http://www.cartoonstock.com/directory/p/probability.asp 11

12. Frequentist Interpretation 12

13. Disjoint 13

14. Disjoint Events: Example Rolling 2 4-sided dice: Events: A: the event that you roll a 2 on the red die B: the event that you roll a 3 on the white die C: the event that the sum of the dice is 3 D: the event that the sum of the dice is 2 Which of the following are mutually exclusive? 1) A and B2) A and C3) A and D 4) B and C5) B and D6) C and D 7) A, B and C8) B, C and D 14

15. Axioms 2.4 15

16. Theorems 16

17. Example: Legitimate Probabilities of Sample Spaces Row #Outcome#1#2#3#4#5#6 110.062500.1 0.05 212120.062500.1 0.05 313130.062500.1 0.05 414140.062500.1 0.05 521210.06250.250.10.20.1 620.06250.250.10.20.1 723230.06250.250.10.20.1 824240.06250.250.10.20.1 931310.062500.1-0.100.01 1032320.062500.1-0.100.01 1130.062500.1-0.100.01 1234340.062500.1-0.100.01 1341410.062500.10.050.10.05 1442420.062500.10.050.10.05 1543430.062500.10.050.10.05 1640.062500.10.050.10.05 17

18. Example: Equally Likely Roll two 4-sided dice a)Determine the probability that the sum of the dice is 5 b)Determine the probability that the red die is more than the white die. 18

19. Dartboard 19

20. Section 2.3: Theorems Complementation Rule: Th. 2.19: P(A c ) = 1 – P(A) Domination Principle: Th. 2.20: If A  B, the P(A) ≤ P(B). 20

21. Example: Complementation Rule (Th. 2.19) 1)If the bulls eye of the dartboard has a radius of 6.3 mm and the whole dartboard has a radius of 225 mm, what is the probability that I will shoot outside of the bulls eye in an equal likelihood situation? 2)What is the probability that I don’t roll a 20 on a 20-sided die? 21

22. Example: Domination Principle (Th. 2.20) Roll a 4-sided die A = roll a 1B = Roll an odd number a)Without doing any calculations, explain why P(A) ≤ P(B) b)Verify your answers by doing the calculations. 22

23. Inclusion/Exclusion (General Addition Rule) Th. 2.22: For any two events A and B, P(A U B) = P(A) + P(B) – P(A ∩ B) 23

24. Example: General Addition Rule (Inclusion/Exclusion) Select a card at random from a deck of cards. What is the probability that the card is either an Ace or a Heart? 24

25. Example for Inclusion-Exclusion Principle Suppose that 60% of the population read the Journal and Courier (J), 25% of the population read the Exponent (E), 10% of the population read the New York Times (N), 15% read both the J and the E, 5% read both the J and the N, 2% read both E and the N and 1% read all three. If a person from this city is selected at random, what is the probability that she does not read at least one of the newspapers? 25

26. Example 2.26 Consider a student who draws cards from a deck. After he draws the card, he replaces the card and then reshuffles the deck. He stops if he draws the ace of spaces. What is the probability of B k, when the ace of spaces is found for the first time on the k th draw? 26

27. Example 2.27 A student hears ten songs (in a random shuffle mode) on her music player, paying special attention to how many of these songs belong to her favorite type of music. We assume the songs are picked independently of each other and that each song has probability p of being a song of the student’s favorite type. What is P(A j ) where A j is the event that exactly j of the 10 songs are from her favorite type of music? 27

28. Chapter 3: Independent Events http://www.cartoonstock.com/directory/p/probability.asp 28

29. Example: Independence Roll a red 4 sided die and a white 4 sided die. LetA: event that the red die is a 1 B: event that the white die is a 1 C: event that the sum of the two dice is 4 a)Are events A and B independent? b)Are events A and C independent? 29

30. Example: Disjoint and Independent Roll a red 4 sided die and a white 4 sided die. Are each of the following disjoint and/or independent? 1) A: event that the red die is a 1 B: event that the red die is a 2 2) A: event that the red die is a 1 B: event that the white die is a 2 3) A: event that the red die is a 1 B: event that the sum of the two dice is 4 30

31. Example: Pairwise Independence Roll a red 4 sided die and a white 4 sided die. LetA: event that the red die is even B: event that the white die is even C: event that the sum of the two dice is even 1)Show that A, B, and C are pairwise independent. 2)Show that A ∩ B and C are NOT independent. 31

32. Example: Mutual Independence Roll a red 6 sided die and a white 6 sided die. LetD: event that the red die is 1 or 2 or 3 E: event that the white die 4 or 5 or 6 F: event that the sum of the two dice is 5 Show that P(D ∩ E ∩ F) = P(D)P(E)P(F) but D, E and F are NOT (mutually) independent events. 32

33. Example 3.19: Independence A student flips a coin until the tenth head appears. Let A denote the event that at least 3 flips are needed between the 7 th and 8 th heads; let B denote the event that at least 3 flips are needed between the 8 th and 9 th heads. 1)What would be considered the trial? 2)Are A and B independent? 33

34. Example: Independence (cont.) If the probability that a fuse is good in a particular batch of fuses is 0.8 and each fuse is independent of the other fuses, what is the probability that 2 fuses are bad? 34

35. Theorem 3.24: Good before Bad 35

36. Chapter 4: Conditional Probability http://imgs.xkcd.com/comics/conditional_risk.png 36

37. Example 1: Conditional Probability Roll a fair 4 sided die 3 times A = the event that two 1’s are tossed B = the event that the first roll is an 1 C = the event that the second roll is an 1 Find: P(B|A), P(A|B), P(B|C) 37

38. Example 2: Conditional Probability A bus arrives punctually at a bus stop every half hour. Each morning, a commuter named Sarah leaves her house and casually strolls to the bus stop. a) Find the probability that that wait time is at least 10 minutes. b) Find the probability that the wait time is at least 10 minutes given that if someone is waiting there for more than 15 minutes, they will get a ride from a passing car. c) Find the probability that the wait time is at most 10 minutes given that if someone is waiting there for more than 15 minutes, they will get a ride from a passing car. 38

39. Example 3: Conditional Probability 1) From the set of all families with two children, a family is selected at random and is found to have a girl. What is the probability that the other child of the family is a girl? 2) From the set of all families with two children, a child is selected at random and is found to be a girl. What is the probability that the other child of the family is a girl? 39

40. Theorem 4.10: Distributive Laws For any events, A 1, A 2, … 40

41. Example: Conditional Probabilities - Axioms Assume that the probability that there will be more than 1 inch of snow next week given that the temperature rises above 30 o F is 0.3. a) What is the probability that there will be less than 1 inch of snow next week given that the temperature rises above 30 o F? b) Given that the probability that there will be more than 2 inches of rain given that the temperature rises above 30 o F is 0.8 and the probability that there will be more than 2 inches of rain and there will be less than 1 inch of snow given that the temperature rises above 30 o F is 0.6, what is the probability that there will be more than 2 inches of rain or there will be less than 1 inch of snow given that the temperature rises above 30 o F? 41

42. Chapter 5: Bayes’ Theorem (And Additional Applications) http://pactiss.org/2011/11/02/bayesian-inference-homo-bayesianis/ 42

43. Example: Bayes’ Theorem In a bolt factory, 30, 50, and 20% of the production is manufactured by machines I, II, and III, respectively. If 4, 5, and 3% of the output of these respective machines is defective, what is the probability that a randomly selected bolt that is found to be defective is manufactured by machine III? 43

44. Example: Bayes’ Theorem (Monty Hall Problem) This follows the game show ‘Let’s Make a Deal’ which was hosted by Monty Hall for many years. In the game show, there are three doors, behind each of which is one prize. Two of the prizes are worthless and the other one is valuable. A contestant selects one of the doors, following which the game show host (who does know where the valuable prize is), opens one of the remaining two doors to reveal a worthless prize. The host then offers the contestant the opportunity to change his selection. Should the contestant switch doors? 44

45. Example: Bayes’ Theorem (Diagnostic Tests) A diagnostic test for a certain disease has a 99% sensitivity and a 95% specificity. Only 1% of the population has the disease in question. If the diagnostic test reports that a person chosen at random from the population tests positive, what is the probability that the person does, in fact, have the disease? 45

46. Examples: General Multiplication Law 1)A consulting firm is awarded 51% of the contracts it bids on. Suppose that Melissa works for a division of the firm that gets to do 25% of the projects contracted for. If Melissa directs 41% of the projects submitted to her division, what percentage of all bids submitted by the firm will result in contracts for projects directed by Melissa? 2)Supposed that 8 good and 2 defective fuses have been mixed up. To find the defective fuses we need to test them one-by-one, at random. Once we test a fuse, we set it aside. What is the probability that we find both of the defective fuses in exactly three tests? 3)Using Pólya’s Urn, with r red balls, b black balls and c, what is the probability that the first two balls are red and the last ball is black? 46

47. Example: Electrical Components For the following problems, assume that each switch is independently closed or open with probability p and 1 - p, respectively. Note: The answers should include ‘p’. 1)In the figure below, there are 4 switches labeled 1, 2, 3 and 4. If a signal is fed to the input, what is the probability that it is transmitted to the output? 2)If a circuit is composed only of n parallel components, then what is the probability that, at a specified time, the system is working? 47