The document provides examples and explanations of probability concepts involving unions and intersections of sets. Example 1a calculates the probability of rolling a sum of 2 or 3 when rolling two fair dice. It shows that this probability is the sum of the individual probabilities of getting a sum of 2 and a sum of 3, since these are mutually exclusive events. Example 1b similarly calculates the probability of an even sum or a sum greater than 4 when rolling two dice.

1. Section 7-2
Addition Counting Principles

2. Warm-up
How many positive integers less than or equal to 1000
satisfy each condition?
a. Divisible by 5? b. Divisible by 7?
Divisible by 5 or 7?

3. Warm-up
How many positive integers less than or equal to 1000
satisfy each condition?
a. Divisible by 5? b. Divisible by 7?
200
Divisible by 5 or 7?

4. Warm-up
How many positive integers less than or equal to 1000
satisfy each condition?
a. Divisible by 5? b. Divisible by 7?
200 142
Divisible by 5 or 7?

5. Warm-up
How many positive integers less than or equal to 1000
satisfy each condition?
a. Divisible by 5? b. Divisible by 7?
200 142
Divisible by 5 or 7?
314

6. Union:

7. Union: Values that are in one set or another; does not
need to be part of both

8. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B

9. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B
Disjoint/Mutually Exclusive:

10. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B
Disjoint/Mutually Exclusive: A situation where two or
more sets have nothing in common

11. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B
Disjoint/Mutually Exclusive: A situation where two or
more sets have nothing in common
i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t
have them both happen at the same time

12. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B
Disjoint/Mutually Exclusive: A situation where two or
more sets have nothing in common
i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t
have them both happen at the same time
Intersection:

13. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B
Disjoint/Mutually Exclusive: A situation where two or
more sets have nothing in common
i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t
have them both happen at the same time
Intersection: Values that are shared by two or more sets

14. Union: Values that are in one set or another; does not
need to be part of both
Notation: A  B
Disjoint/Mutually Exclusive: A situation where two or
more sets have nothing in common
i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t
have them both happen at the same time
Intersection: Values that are shared by two or more sets
Notation: A  B

15. Addition Counting Principle (Mutually Exclusive Form):

16. Addition Counting Principle (Mutually Exclusive Form):
If two finite sets A and B are mutually exclusive, then
N( A  B) = N( A) + N(B)

17. Addition Counting Principle (Mutually Exclusive Form):
If two finite sets A and B are mutually exclusive, then
N( A  B) = N( A) + N(B)
Theorem (Probability of the Union of Mutually Exclusive Events):

18. Addition Counting Principle (Mutually Exclusive Form):
If two finite sets A and B are mutually exclusive, then
N( A  B) = N( A) + N(B)
Theorem (Probability of the Union of Mutually Exclusive Events):
If A and B are mutually exclusive events in the same finite
sample space, then
P( A  B) = P( A) + P(B)

19. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?

20. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

21. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

22. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

23. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

24. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

25. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6 = 1
36
+ 36
2
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

26. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6 = 1
36
+ 36 =
2 3
36
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

27. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6 = 1
36
+ 36 =
2 3
36
= 12
1
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

28. Example 1
a. If two fair dice are tossed, what is the probability that
the sum is 2 or 3?
1,1 1,2 1,3 1,4 1,5 1,6 P(sum of 2 or 3)
2,1 2,2 2,3 2,4 2,5 2,6 = P(sum of 2) + P(sum of 3)
3,1 3,2 3,3 3,4 3,5 3,6 = 1
36
+ 36 =
2 3
36
= 12
1
4,1 4,2 4,3 4,4 4,5 4,6 There is an 8 1/3 %
5,1 5,2 5,3 5,4 5,5 5,6 chance of rolling a sum
6,1 6,2 6,3 6,4 6,5 6,6 of 2 or 3

29. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?

30. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

31. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

32. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

33. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

34. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

35. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
= 36
18
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

36. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
= 36 + 36
18 30
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

37. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
= 36 + 36 − 36
18 30 14
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

38. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
= 36 + 36 − 36 = 36
18 30 14 34
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

39. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
= 36 + 36 − 36 = 36 = 18
18 30 14 34 17
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

40. Example 1
b. If two dice are tossed, what is the probability that the
sum will be even or greater than 4?
P(sum is even or > 4)
1,1 1,2 1,3 1,4 1,5 1,6
= P(sum is even) + P(sum > 4)
2,1 2,2 2,3 2,4 2,5 2,6
= 36 + 36 − 36 = 36 = 18
18 30 14 34 17
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6 There is a 94 4/9 %
5,1 5,2 5,3 5,4 5,5 5,6 chance of rolling an
even sum or a sum
6,1 6,2 6,3 6,4 6,5 6,6 greater than 4

41. Addition Counting Principle (General Form):

42. Addition Counting Principle (General Form):
For any finite sets A and B,
N( A  B) = N( A) + N(B) − N( A  B)

43. Addition Counting Principle (General Form):
For any finite sets A and B,
N( A  B) = N( A) + N(B) − N( A  B)
Theorem (Probability of a Union of Events General Form):

44. Addition Counting Principle (General Form):
For any finite sets A and B,
N( A  B) = N( A) + N(B) − N( A  B)
Theorem (Probability of a Union of Events General Form):
If A and B are any events in the same finite sample space,
then
P( A or B) = P( A  B) = P( A) + P(B) − P( A  B)

45. Example 2
Thirteen of the 50 states include territory that lies west of
the continental divide. Forty-two states include territory
that lies east of the continental divide. Is this possible?
Explain.

46. Example 2
Thirteen of the 50 states include territory that lies west of
the continental divide. Forty-two states include territory
that lies east of the continental divide. Is this possible?
Explain.
Of course this is possible! This just means that some states
have the continental divide running right though them!

47. Example 2
Thirteen of the 50 states include territory that lies west of
the continental divide. Forty-two states include territory
that lies east of the continental divide. Is this possible?
Explain.
Of course this is possible! This just means that some states
have the continental divide running right though them!
Bonus: Which states would these be? The FIRST person
to post the correct answer AFTER 7 PM on the wiki
under section 7-2 will earn some bonus points!

48. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?

49. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?
HHH HTT
HHT THT
HTH TTH
THH TTT

50. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?
HHH HTT
HHT THT
HTH TTH
THH TTT

51. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?
P(Not all 3 same)
HHH HTT
HHT THT
HTH TTH
THH TTT

52. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?
P(Not all 3 same)
HHH HTT = 6
8
HHT THT
HTH TTH
THH TTT

53. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?
P(Not all 3 same)
HHH HTT = 6
8
= 3
4
HHT THT
HTH TTH
THH TTT

54. Example 3
Three fair coins are tossed. What is the probability that not
all of the coins show the same face?
P(Not all 3 same)
HHH HTT = 6
8
= 3
4
HHT THT
HTH TTH There is a 75%
THH TTT chance of
getting not all 3
coins the same

55. Complementary Events

56. Complementary Events
When you have events whose union takes up the entire
sample space, but the events are mutually exclusive

57. Complementary Events
When you have events whose union takes up the entire
sample space, but the events are mutually exclusive
The complement of R is not R

58. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.

59. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

60. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

61. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.
1,1 1,2 1,3 1,4 1,5 1,6 P(not 6)
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

62. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.
1,1 1,2 1,3 1,4 1,5 1,6 P(not 6) = 31
36
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

63. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.
1,1 1,2 1,3 1,4 1,5 1,6 P(not 6) = 31
36
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6 There is an 86 1/9 %
chance that the sum
4,1 4,2 4,3 4,4 4,5 4,6 will not be 6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6

64. Example 4
Two dice are tossed. Find the probability that their sum is
not 6.
1,1 1,2 1,3 1,4 1,5 1,6 P(not 6) = 31
36
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6 There is an 86 1/9 %
chance that the sum
4,1 4,2 4,3 4,4 4,5 4,6 will not be 6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6 1− P(6) =1− 5
36
= 31
36

65. Theorem (Probability
of Complements)

66. Theorem (Probability
of Complements)
In any event E, the complement of E is
P(not E) =1− P(E)

67. Homework

68. Homework
p. 437 #1-24