1. Chapter 4 – Probability Concepts
Introduction to Business Statistics, 6e
Kvanli, Pavur, Keeling
Chapter 4 – Probability Concepts
Slides prepared by Jeff Heyl, Lincoln University
©2003 South-Western/Thomson Learning™

2. Events and Probability
An activity for which the outcome is uncertain is an experiment
An event consists of one more possible outcomes of the experiment

3. This assumes all n possible outcomes have an equal chance of occurring
Classical Definition
P(A) = probability that event A occurs
The probability of event m occurring in n outcomes
P(A) = m/n
This assumes all n possible outcomes have an equal chance of occurring

4. Relative Frequency Approach
P(A) = m/n
Observe an experiment n times and count the number of times event A occurs, m

5. Subjective Probability
A measure (between 0 and 1) of your belief that a particular event will occur

6. Datacomp Survey Age (Years) <30 30 - 45 >45
< >45
Sex (U) (B) (O) Total
Male (M)
Female (F)
Total
M = a male is selected
F = a female is selected
U = the person selected is under 30
B = the person selected is between 30 and 45
O = the person selected is over 45
Table 4.1

7. Marginal Probability Marginal Probability the probability of a single
event used to define the contingency table
P(M) = 120/200 = .6
P(F) = 80/200 = .4
P(U) = .5
P(B) = .25
P(O) = .25

8. Complement of an Event
The complement of an event A is the event that A does not occur: A
P(A) + P(A) = 1
P(M) = 1 - P(M) = .4

9. The probability of the occurrence of two events at the same time
Joint Probability
The probability of the occurrence of two events at the same time
The probability of selecting a person who is a female and under 30
P(F and U) = 40/200 = .2

10. Union of Events
The union of events is the probability of either event A or event B occurring:
P(A or B)
The probability of selecting a person who is male or under 30
P(M or U) = ( ) / 200 = .8

11. Conditional Probability
Whenever you are given information and are asked to find a probability based on this information, the result is a conditional probability.
P(A | B)

12. Independent Events
If the P(A) = P(A | B) then event A is said to be independent of event B
P(M) = P(M | U) = .6
Thus event M is independent of event U

13. Independent Events Events A and B are independent if and only if:
1. P(A | B) = P(A) (assuming P(B) ≠ 0), or
2. P(B | A) = P(B) (assuming P(A) ≠ 0), or
3. P(A and B) = P(A) • P(B)

14. Mutually Exclusive Events
If an event can not occur when another event has occurred the two events are said to be mutually exclusive
Selecting a male and a female are mutually exclusive events
P(M and F) = 0

15. Venn Diagrams A B
Figure 4.1

16. Venn Diagram for P(A) = .4 A .4 .6
Figure 4.2

17. Venn Diagram for P(A and B)
Figure 4.3

18. Venn Diagram for P(A or B)
Figure 4.4

19. Venn Diagram of Mutually Exclusive EventsB
.25
Figure 4.5

20. Venn Diagram for P(A or B) and P(A and B)
Figure 4.6

21. Additive Probability Rules
General Additive Rule
P(A or B) = P(A) + P(B) - P(A and B)
Special Additive Rule
If A and B are mutually exclusive then:
P(A or B) = P(A) + P(B)

22. Venn Diagram for Example 4.2
= P(Q or H)
= P(Q and H) Q H
1/52
Figure 4.7

23. Venn Diagram for Conditional Probability
= P(A and B)
= P(B)
Figure 4.8

24. Conditional Probabilities
General Conditional Probability Rule
P(A | B) = (P(B) ≠ 0)
and
P(B | A) = (P(A) ≠ 0)
P(A and B)
P(B)
P(A)

25. Conditional Probabilities
Special Conditional Probability Rule
If A and B are independent then:
P(A | B) = P(A)
P(B | A) = P(B)
P(A and B) = P(A) • P(B)

26. Multiplicative Rule For any two events A and B:
P(A and B) = P(A | B) • P(B)
= P(B | A) • P(A)
For any two independent events A and B:
P(A and B) = P(A) • P(B)

27. Contingency Table
Figure 4.9

28. Venn Diagram for Example 4.5.2 .3
P(M and E) = .1
P(M and E)
Figure 4.10

29. General Tree Diagram B E1 E2 En .
Figure 4.11

30. Rules for Tree Diagrams
Rule #1: The probability of the event on the right side (say, event B) of the tree is equal to the sum of the paths; that is, all probabilities along a path leading to event B are multiplied, and then summed over all paths leading to B

31. Rules for Tree Diagrams
Rule #2: The posterior probability for the i th path is
P(Ei | B) =
where the “sum of paths” is found using Rule #1
i th path
sum of paths

32. Mutually Exclusive Events
Events A, B, and C are pairwise mutually exclusive if no two events can occur simultaneously
P(A or B or C) = P(A) + P(B) + P(C) A B C
Figure 4.12

33. P(A and B and C) = P(A) • P(B) • P(C)
Independent Events
Events A, B, and C are independent if all of the following are true:
P(A and B) = P(A) • P(B)
P(A and C) = P(A) • P(C)
P(B and C) = P(B) • P(C)
P(A and B and C) = P(A) • P(B) • P(C)

34. Counting Rules
Counting Rules determine the number of outcomes that exist for a certain broad range of experiments.
Filling Slots
Permutations
Combinations

35. Filling Slots Use counting rule 1 to fill k different slots Let:
n1 = the number of ways to filling the first slot
n2 = the number of ways to filling the second
slot after the first slot is filled 
nk = the number of ways to filling the kth slot
after slots 1 though k - 1 are filled
The number of ways of filling all k slots is:
n1  n2  n3  …  nk

36. Permutations
Permutations is the counting situation in which samples are taken without replacement and where order of selection is important
nPk = = (n)(n - 1) ... (n - k + 1) n!
(n - k)!

37. Combinations
Combinations is the counting situation in which samples are taken without replacement and where order of selection is not important
nCk = = ·
nPk k! n!
k!(n - k)!