1. Statistics
Chapter 3: Probability

2. McClave: Statistics, 11th ed. Chapter 3: Probability
Where We’ve Been
Making Inferences about a Population Based on a Sample
Graphical and Numerical Descriptive Measures for Qualitative and Quantitative Data
McClave: Statistics, 11th ed. Chapter 3: Probability

3. McClave: Statistics, 11th ed. Chapter 3: Probability
Where We’re Going
Probability as a Measure of Uncertainty
Basic Rules for Finding Probabilities
Probability as a Measure of Reliability for an Inference
McClave: Statistics, 11th ed. Chapter 3: Probability

4. 3.1: Events, Sample Spaces and Probability
An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty.
McClave: Statistics, 11th ed. Chapter 3: Probability

5. 3.1: Events, Sample Spaces and Probability
A sample point is the most basic outcome of an experiment.
An Ace
A four
A Head
McClave: Statistics, 11th ed. Chapter 3: Probability

6. 3.1: Events, Sample Spaces and Probability
A sample space of an experiment is the collection of all sample points.
Roll a single die:
S: {1, 2, 3, 4, 5, 6}
McClave: Statistics, 11th ed. Chapter 3: Probability

7. 3.1: Events, Sample Spaces and Probability
Sample points and sample spaces are often represented with Venn diagrams.
McClave: Statistics, 11th ed. Chapter 3: Probability

8. 3.1: Events, Sample Spaces and Probability
Probability Rules for Sample Points
All probabilities must be between 0 and 1.
The probabilities of all the sample points must sum to 1.
McClave: Statistics, 11th ed. Chapter 3: Probability

9. 3.1: Events, Sample Spaces and Probability
Probability Rules for Sample Points
All probabilities must be between 0 and 1
The probabilities of all the sample points must sum to 1
0 indicates an impossible outcome and 1 a certain outcome.
Something has to happen.
McClave: Statistics, 11th ed. Chapter 3: Probability

10. 3.1: Events, Sample Spaces and Probability
An event is a specific collection of sample points:
Event A: Observe an even number.
McClave: Statistics, 11th ed. Chapter 3: Probability

11. 3.1: Events, Sample Spaces and Probability
The probability of an event is the sum of the probabilities of the sample points in the sample space for the event.
Event A: Observe an even number.
P(A) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
McClave: Statistics, 11th ed. Chapter 3: Probability

12. 3.1: Events, Sample Spaces and Probability
Calculating Probabilities for Events
Define the experiment.
List the sample points.
Assign probabilities to sample points.
Collect all sample points in the event of interest.
The sum of the sample point probabilities is the event probability.
McClave: Statistics, 11th ed. Chapter 3: Probability

13. 3.1: Events, Sample Spaces and Probability
Calculating Probabilities for Events
Define the experiment
List the sample points
Assign probabilities to sample points
Collect all sample points in the event of interest
The sum of the sample point probabilities is the event probability
If the number of sample points gets too large, we need a way to keep track of how they can be combined for different events.
McClave: Statistics, 11th ed. Chapter 3: Probability

14. 3.1: Events, Sample Spaces and Probability
If a sample of n elements is drawn from a set of N elements (N ≥ n), the number of different samples is
McClave: Statistics, 11th ed. Chapter 3: Probability

15. 3.1: Events, Sample Spaces and Probability
If a sample of 5 elements is drawn from a set of 20 elements, the number of different samples is
McClave: Statistics, 11th ed. Chapter 3: Probability

16. 3.2: Unions and Intersections
McClave: Statistics, 11th ed. Chapter 3: Probability

17. 3.2: Unions and Intersections
McClave: Statistics, 11th ed. Chapter 3: Probability

18. 3.2: Unions and Intersections
A  C
A  B  C B C
McClave: Statistics, 11th ed. Chapter 3: Probability

19. 3.3: Complementary Events
The complement of any event A is the event that A does not occur, AC.
A: {Toss an even number}
AC: {Toss an odd number}
B: {Toss a number ≤ 3}
BC: {Toss a number ≥ 4}
A  B = {1,2,3,4,6}
[A  B]C = {5}
(Neither A nor B occur)
McClave: Statistics, 11th ed. Chapter 3: Probability

20. 3.3: Complementary Events
McClave: Statistics, 11th ed. Chapter 3: Probability

21. 3.3: Complementary Events
A: {At least one head on two coin flips}
AC: {No heads}
McClave: Statistics, 11th ed. Chapter 3: Probability

22. 3.4: The Additive Rule and Mutually Exclusive Events
The probability of the union of events A and B is the sum of the probabilities of A and B minus the probability of the intersection of A and B:
McClave: Statistics, 11th ed. Chapter 3: Probability

23. 3.4: The Additive Rule and Mutually Exclusive Events
At a particular hospital, the probability of a patient having surgery (Event A) is .12, of an obstetric treatment (Event B) .16, and of both .02.
What is the probability that a patient will have either treatment?
McClave: Statistics, 11th ed. Chapter 3: Probability

24. 3.4: The Additive Rule and Mutually Exclusive Events
Events A and B are mutually exclusive if A  B contains no sample points.
McClave: Statistics, 11th ed. Chapter 3: Probability

25. 3.4: The Additive Rule and Mutually Exclusive Events
If A and B are mutually exclusive,
McClave: Statistics, 11th ed. Chapter 3: Probability

26. 3.5: Conditional Probability
Additional information or other events occurring may have an impact on the probability of an event.
McClave: Statistics, 11th ed. Chapter 3: Probability

27. 3.5: Conditional Probability
Additional information may have an impact on the probability of an event.
P(Rolling a 6) is one-sixth (unconditionally).
If we know an even number was rolled, the probability of a 6 goes up to one-third.
McClave: Statistics, 11th ed. Chapter 3: Probability

28. 3.5: Conditional Probability
The sample space is reduced to only the conditioning event.
To find P(A), once we know B has occurred (i.e., given B), we ignore BC (including the A region within BC). B BC A A
McClave: Statistics, 11th ed. Chapter 3: Probability

29. 3.5: Conditional ProbabilityBC A A
McClave: Statistics, 11th ed. Chapter 3: Probability

30. 3.5: Conditional Probability
McClave: Statistics, 11th ed. Chapter 3: Probability

31. 3.5: Conditional Probability
55% of sampled executives had cheated at golf (event A).
P(A) = .55
20% of sampled executives had cheated at golf and lied in business (event B).
P(A  B) = .20
What is the probability that an executive had lied in business, given s/he had cheated in golf, P(B|A)?
McClave: Statistics, 11th ed. Chapter 3: Probability

32. 3.5: Conditional Probability
P(A) = .55
P(A  B) = .20
What is P(B|A)?
McClave: Statistics, 11th ed. Chapter 3: Probability

33. 3.6: The Multiplicative Rule and Independent Events
The conditional probability formula can be rearranged into the Multiplicative Rule of Probability to find joint probability.
McClave: Statistics, 11th ed. Chapter 3: Probability

34. 3.6: The Multiplicative Rule and Independent Events
Assume three of ten workers give illegal deductions
Event A: {First worker selected gives an illegal deduction}
Event B: {Second worker selected gives an illegal deduction}
P(A) = P(B) = = .3
P(B|A) has only nine sample points, and two targeted workers, since we selected one of the targeted workers in the first round:
P(B|A) = 1/9 + 1/9 = = .22
The probability that both of the first two workers selected will have given illegal deductions
P(AB) = P(B|A)P(A) = .(3) (.22) = .066
McClave: Statistics, 11th ed. Chapter 3: Probability

35. 3.6: The Multiplicative Rule and Independent Events
McClave: Statistics, 11th ed. Chapter 3: Probability

36. 3.6: The Multiplicative Rule and Independent Events
10 Workers
Illegal
Deductions
P = .3
P = .3 x.22
= .066
No Illegal Deductions
P = .3 x.78
= .234
No Illegal
P = .7
P = .7 x.33
= .231
P = .7 x .67
= .469
A Tree Diagram
First selected worker
Second selected worker
McClave: Statistics, 11th ed. Chapter 3: Probability

37. 3.6: The Multiplicative Rule and Independent Events
P(A|B) ≠ P(A)
P(B|A) ≠ P(B)
Independent Events
P(A|B) = P(A)
P(B|A) = P(B)
 Studying stats 40 hours per week
Working 40 hours per week
Having blue eyes
McClave: Statistics, 11th ed. Chapter 3: Probability

38. 3.6: The Multiplicative Rule and Independent Events
P(A|B) ≠ P(A)
P(B|A) ≠ P(B)
Independent Events
P(A|B) = P(A)
P(B|A) = P(B)
Mutually exclusive events are dependent: P(B|A) = 0
Since P(B|A) = P(B),
P(AB) = P(A)P(B|A) = P(A)P(B)
McClave: Statistics, 11th ed. Chapter 3: Probability

39. McClave: Statistics, 11th ed. Chapter 3: Probability
3.7: Random Sampling
If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a (simple) random sample.
McClave: Statistics, 11th ed. Chapter 3: Probability

40. McClave: Statistics, 11th ed. Chapter 3: Probability
3.7: Random Sampling
How many five-card poker hands can be dealt from a standard 52-card deck?
Use the combination rule:
McClave: Statistics, 11th ed. Chapter 3: Probability

41. McClave: Statistics, 11th ed. Chapter 3: Probability
3.7: Random Sampling
Random samples can be generated by
Mixing up the elements and drawing by hand, say, out of a hat (for small populations)
Random number generators
Random number tables
Table I, App. B
Random sample/number commands on software
McClave: Statistics, 11th ed. Chapter 3: Probability

42. 3.8: Some Additional Counting Rules
Situation
Number of Different Results
Multiplicative Rule
Draw one element from each of k sets, sized n1, n2, n3, … nk
Permutations Rule
Draw n elements, arranged in a distinct order, from a set of N elements
Partitions Rule
Partition N elements into k groups, sized n1, n2, n3, … nk (ni=N)
McClave: Statistics, 11th ed. Chapter 3: Probability

43. 3.8: Some Additional Counting Rules
Multiplicative Rule
Assume one professor from each of the six departments in a division will be selected for a special committee. The various departments have four, seven, six, eight, six and five professors eligible. How many different committees, X*, could be formed?
McClave: Statistics, 11th ed. Chapter 3: Probability

44. 3.8: Some Additional Counting Rules
Permutations
If there are eight horses entered in a race, how many different win, place and show possibilities are there?
McClave: Statistics, 11th ed. Chapter 3: Probability

45. 3.8: Some Additional Counting Rules
Partitions Rule
The technical director of a theatre has twenty stagehands. She needs eight electricians, ten carpenters and two props people. How many different allocations of stagehands, Y*, can there be?
McClave: Statistics, 11th ed. Chapter 3: Probability

46. McClave: Statistics, 11th ed. Chapter 3: Probability
3.8: Bayes’s Rule
Given k mutually exclusive and exhaustive events B1, B2,… Bk, and an observed event A, then
McClave: Statistics, 11th ed. Chapter 3: Probability

47. McClave: Statistics, 11th ed. Chapter 3: Probability
3.8: Bayes’s Rule
Suppose the events B1, B2, and B3, are mutually exclusive and complementary events with P(B1) = .2, P(B2) = .15 and P(B3) = Another event A has these conditional probabilities: P(A|B1) = .4, P(A|B2) = .25 and P(A|B3) = .6.
What is P(B1|A)?
McClave: Statistics, 11th ed. Chapter 3: Probability