1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman
Multiplication Rule
Sections 3-4 & 3-5
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

2. Finding the Probability of Two or More Selections
Multiple selections
Multiplication Rule

3. Example:
Tom and Fred are playing a game. One of them is asked to pick a card from a pocket, that contains 5 cards marked a, b, c, d, e.
Q: What is the probability of the event
A = { Tom picks the “c” card }

4. FIGURE Tree Diagram of Test Answers

5. T F FIGURE Tree Diagram of the events T & a T & b T & c T & d T & e
F & a
F & b
F & c
F & d
F & e T F

6. T F FIGURE Tree Diagram of Test Answers T & a T & b T & c T & d T & e
F & a
F & b
F & c
F & d
F & e T F

7. P(T) = T F FIGURE Tree Diagram of Test Answers T & a T & b T & c T & d
F & a
F & b
F & c
F & d
F & e T F 1 2
P(T) =

8. P(T) = P(c) = T F FIGURE Tree Diagram of Test Answers T & a T & b
T & c
T & d
T & e
F & a
F & b
F & c
F & d
F & e T F 1 2 1 5
P(T) = P(c) =

9. P(T) = P(c) = P(T and c) = T F FIGURE Tree Diagram of Test Answersb c d e
T & a
T & b
T & c
T & d
T & e
F & a
F & b
F & c
F & d
F & e T F 1 2 1 5 1 10
P(T) = P(c) = P(T and c) =

10. P (Tom and c card)

11. P (Tom and c card) = P (T and c)

12. P (Tom and c card) = P (T and c)1 10 1 1 2 5

13. P (Tom and c card) = P (T and c)1 10 1 1 = • 2 5
Multiplication
Rule

14. INDEPENDENT EVENTS = • 1 1 1 10 2 5 P (Tom and c card) = P (T and c)
Multiplication
Rule
INDEPENDENT EVENTS

15. Definitions Independent Events Dependent Events
Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other.
Dependent Events
If A and B are not independent, they are said to be dependent.

16. Notation for Multiplication Rule
P(B | A) represents the probability of B occurring after it is assumed that event A has already occurred (read B | A as “B given A”).

17. Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.)
P(Ace on first card) =
P(King/Ace) =
P(drawing Ace, then a King) = • = 4 52 4 51 4 52 4 51 16
.00603
2652
DEPENDENT EVENTS

18. Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.)
P(Ace on first card) = 4 52

19. Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.)
P(Ace on first card) =
P(King Ace) = 4 52 4 51

20. Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.)
P(Ace on first card) =
P(King Ace) =
P(drawing Ace, then a King) = • = 4 52 4 51 4 52 4 51 16
2652

21. Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.)
P(Ace on first card) =
P(King Ace) =
P(drawing Ace, then a King) = • = 4 52 4 51 4 52 4 51 16
2652
DEPENDENT EVENTS

22. Formal Multiplication Rule
P(A and B) = P(A) • P(B) if A and B are independent

23. Formal Multiplication Rule
P(A and B) = P(A) • P(B) if A and B are independent
P(A and B) = P(A) • P(B|A) if A and B are dependent

24. P(A and B) = P(A) • P(B | A)
Figure Applying the Multiplication Rule
P(A or B)
Multiplication Rule
Are
A and B
independent ?
Yes
P(A and B) = P(A) • P(B) No
P(A and B) = P(A) • P(B | A)

25. Intuitive Multiplication
When finding the probability that event A occurs on one trial and B occurs on the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

26. Small Samples from Large Populations
If small sample is drawn from large population (if n £ 5% of N), you can treat the events as independent.

27. Conditional Probability
Definition
The conditional probability of B given A is the probability of event B occurring, given that A has already occurred.

28. Conditional Probability
Dependent Events
P(A and B) = P(A) • P(B|A)

29. Conditional Probability
Dependent Events
P(A and B) = P(A) • P(B|A)
Formal

30. Conditional Probability
Dependent Events
P(A and B) = P(A) • P(B|A)
Formal
P(B|A) =
P(A and B)
P(A)

31. Conditional Probability
Dependent Events
P(A and B) = P(A) • P(B|A)
Formal
P(B|A) =
Intuitive
P(A and B)
P(A)

32. Conditional Probability
Dependent Events
P(A and B) = P(A) • P(B|A)
Formal
P(B|A) =
Intuitive
The conditional probability of B given A can be
found by assuming the event A has occurred and,
operating under that assumption, calculating the
probability that event B will occur.
P(A and B)
P(A)

33. Probability of ‘At Least One’

34. Probability of ‘At Least One’
‘At least one’ is equivalent to one or more.

35. Probability of ‘At Least One’
‘At least one’ is equivalent to one or more.
The complement of getting at least one item of a particular type is that you get no items of that type.

36. Probability of ‘At Least One’
‘At least one’ is equivalent to one or more.
The complement of getting at least one item of a particular type is that you get no items of that type.
If P(A) = P(getting at least one), then

37. Probability of ‘At Least One’
‘At least one’ is equivalent to one or more.
The complement of getting at least one item of a particular type is that you get no items of that type.
If P(A) = P(getting at least one), then
P(A) = 1 – P(A)

38. Probability of ‘At Least One’
‘At least one’ is equivalent to one or more.
The complement of getting at least one item of a particular type is that you get no items of that type.
If P(A) = P(getting at least one), then
P(A) = 1 – P(A)
where P(A) is P(getting none)