1. Probability

2. An experiment is any process that allows researchers to obtain observations and which leads to a single outcome which cannot be predicted with certainty

3. Probability An experiment is any process that allows researchers to obtain observations and which leads to a single outcome which cannot be predicted with certainty Example: A coin is tossed…possible outcomes are H, T. Example: Two coins are tossed…possible outcomes are HH, HT, TH, TT.

4. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Tree diagram for the coin-tossing experiment

5. Probability An outcome is also called a sample point. The collection of all possible sample points is called the sample space..

6. Probability An outcome is also called a sample point. The collection of all possible sample points is called the sample space. Example: Roll a die and observe the result. The sample space consists of the set S = {1,2,3,4,5,6}

7. Example When we roll two dice the sample space is all 36 possible pairs of rolls 1,1 1,2 1,31,4 1,5 1,6 2,1 2,2 2,32,4 2,5 2,6 3,13,2 3,33,4 3,5 3,6 4,14,2 4,34,4 4,5 4,6 5,15,2 5,35,4 5,5 5,6 6,16,2 6,36,4 6,5 6,6

8. The probability of a single outcome If an experiment is repeated many times, and if a particular outcome is known or observed to occur, for example, 1/6 th of the time, we say that the probability of that outcome is 1/6. Sometimes one can compute probabilities exactly. E.g., if a bowl contains 5 red chips and 3 Blue chips and if a chip is selected at random, then the probability a blue chip is selected is 3/8. In general, it may be difficult to determine the probability of an outcome. (for example, what is the probability of rain tomorrow).

9. In general, the probabilty that a particular outcome occurs is defined to be the fraction of times which that outcome occurs in the long run.

10. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Proportion of heads in N tosses of a coin

11. Equi-probability sample spaces A sample space S in which all sample points are equally likely to occur is called an equi-probability space. Definition: For an equi-probability space, the probability of a particular outcome is 1/n, where n is the number of sample points in S.

12. Example A coin is tossed and a die is rolled. What is the probability that the outcome is H6?

13. Example A coin is tossed and a die is rolled. What is the probability that the outcome is H6? Answer: S={H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} Therefore P(H6) = 1/12.

14. Events It turns out that we are usually interested NOT in a single outcome, but rather a collection of outcomes forming a subset of the sample space. Definition: An event is a specific collection of sample points.

15. Sample Space Event Simple Events (all the red dots) We often represent the sample space with a Venn Diagram.

16. Probability of an event NOTE: True in general NOT just for equi-probability case

17. NOTE For an equi-probability sample space, the probability of an event E occurring is simply the number of sample points in E divided by the total number of sample points in the sample space S.

18. Sample Space Example: Roll two dice. What is the probability of rolling a 9? 1,1 1,2 1,31,4 1,5 1,6 2,1 2,2 2,32,4 2,5 2,6 3,13,2 3,33,4 3,5 3,6 4,14,2 4,34,4 4,5 4,6 5,15,2 5,35,4 5,5 5,6 6,16,2 6,36,4 6,5 6,6

19. Sample Space Example: Roll two dice. What is the probability of rolling a 9?

20. Sample Space Example: Roll two dice. What is the probability of rolling a 9?

21. Properties of Probability

22. Union

23. The union of events A and B is the event that A or B (or both) occur. A B A or B

24. Intersection

25. The intersection of events A and B is the event that both A and B occur. A B A and B

26. Compliment

27. The compliment of an event A is the event that A does not occur. ACAC

28. Compliment Rule

29. Addition Rule P(A or B)= P(A) + P(B) - P(A and B) “Proof”: Consider the case of an equi-probability space. To find P(A or B) we must count all the points in the set We can do that by adding together all of the points in A plus all the points in B. But this will count all of those points which lie in BOTH A and B twice so we must subtract the number of points in the intersection Letting N be the number of points in the entire sample space and using the notation that n(A) denotes the number of points in a set A, we obtain

30. Example For an experiment of randomly selecting one card from a deck of 52 cards, let A=event the card selected is the King of Hearts B=event the card selected is a King C=event the card selected is a Heart D=event the card selected is a face card. Find: a)P(D C ) b) P(B and C) c)P(B or C) d) P(C and D) e)P(A or B)f) P(B)

31. Example For an experiment of randomly selecting one card from a deck of 52 cards, let A=event the card selected is the King of Hearts B=event the card selected is a King C=event the card selected is a Heart D=event the card selected is a face card. Find: a)P(D C ) =40/52 b) P(B and C)= 1/52 c)P(B or C)=16/52d) P(C and D)=3/52 e)P(A or B)=4/52f) P(B)=4/52

32. Mutually Exclusive Events Two events are mutually exclusive if P (A and B) = 0. In terms of sets, this means that the sets A and B to do not overlap. For mutually exclusive sets, we have a simplified addition rule:

33. Mutually Exclusive Two events are mutually exclusive if P (A and B) = 0. Suppose P (E) =.3, P (F) =.5, and E and F are mutually exclusive. Find: P(E and F)= P(E or F)= P(E C )= P(F C )= P((E or F) C )= P((E and F) C )=

34. Mutually Exclusive Two events are mutually exclusive if P (A and B) = 0. Suppose P (E) =.3, P (F) =.5, and E and F are mutually exclusive. Find: P(E and F) = 0 P(E or F) = 0.8 P(E C ) = 0.7 P(F C ) = 0.5 P((E or F) C )=0.2 P((E and F) C )=1

35. Example A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse? Frequency Black7 Brown4 White1 Total12

36. Example A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse? FrequencyProbability Black77/12 Brown44/12 White11/12 Total121

37. Example A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse? P (Black or White) =

38. Example A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse? P (Black or White) = P(Black) +P(White)

39. Example A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse? P (Black or White) = P(Black) +P(White) = 7/12 + 1/12 = 8/12

40. Example In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty. a) Find the probability that a computer set purchased has one of the two problems b) Neither c) Just a monitor problem

41. Example In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty. a) Find the probability that a computer set purchased has one of the two problems. (5.8%) b) Neither c) Just a monitor problem

42. Example In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty. a) Find the probability that a computer set purchased has one of the two problems. (5.8%) b) Neither (94.2%) c) Just a monitor problem (1.8%)

43. Example In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty. a) Find the probability that a computer set purchased has one of the two problems. (5.8%) b) Neither (94.2%) c) Just a monitor problem (1.8%)

44. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) P(E C or F C ) P(E C and F C )P(E C and F)

45. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C ) P(E C and F C )P(E C and F)

46. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )P(E C and F)

47. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4P(E C and F)

48. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4P(E C and F)= 0.2

49. Review Probabilities –Definitions of experiment, event, simple event, sample space, probabilities, intersection, union compliment –Finding Probabilities –Drawing Venn Diagrams –If A and B are two events then P(A or B) = P(A) + P(B) - P(A and B), P(not A) = 1 - P(A). –Two events A and B are mutually exclusive if P(A and B) = 0.