Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds
1. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Math 1300 Finite Mathematics
Section 8-2: Union, Intersection, and Complement of Events;
Odds
Jason Aubrey
Department of Mathematics
University of Missouri
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
2. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
In this section, we will develop the rules of probability for
compound events (more than one simple event) and will
discuss probabilities involving the union of events as well as
intersection of two events.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
3. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
If A and B are two events in a sample space S, then the union
of A and B, denoted by A ∪ B, and the intersection of A and B,
denoted by A ∩ B, are defined as follows:
Definition (Union: A ∪ B)
A ∪ B = {e ∈ S|e ∈ A or e ∈ B}
A B
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
4. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
If A and B are two events in a sample space S, then the union
of A and B, denoted by A ∪ B, and the intersection of A and B,
denoted by A ∩ B, are defined as follows:
Definition (Union: A ∪ B)
A ∪ B = {e ∈ S|e ∈ A or e ∈ B}
A B
../images/stackedlogo-bw-
The event “A or B” is defined as the set A ∪ B.
Jason Aubrey Math 1300 Finite Mathematics
5. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Definition (Intersection: A ∩ B)
A ∩ B = {e ∈ S|e ∈ A and e ∈ B}
A B
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
6. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Definition (Intersection: A ∩ B)
A ∩ B = {e ∈ S|e ∈ A and e ∈ B}
A B
The event “A and B” is defined as the set A ∩ B. ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
7. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Consider the sample space of equally likely events
for the rolling of a single fair die: S = {1, 2, 3, 4, 5, 6}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
8. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Consider the sample space of equally likely events
for the rolling of a single fair die: S = {1, 2, 3, 4, 5, 6}
(a) What is the probability of rolling an odd number and a prime
number?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
9. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Consider the sample space of equally likely events
for the rolling of a single fair die: S = {1, 2, 3, 4, 5, 6}
(a) What is the probability of rolling an odd number and a prime
number?
Let A be the event “an odd number is rolled”. Let B be the event
“a prime number is rolled”.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
10. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Consider the sample space of equally likely events
for the rolling of a single fair die: S = {1, 2, 3, 4, 5, 6}
(a) What is the probability of rolling an odd number and a prime
number?
Let A be the event “an odd number is rolled”. Let B be the event
“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,
A ∩ B = {3, 5}.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
11. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Consider the sample space of equally likely events
for the rolling of a single fair die: S = {1, 2, 3, 4, 5, 6}
(a) What is the probability of rolling an odd number and a prime
number?
Let A be the event “an odd number is rolled”. Let B be the event
“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,
A ∩ B = {3, 5}.
By the equally likely assumption,
n(A ∩ B) 2 1
P(A ∩ B) = = =
n(S) 6 3
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
12. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
(b) What is the probability of rolling an odd number or a prime
number?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
13. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
(b) What is the probability of rolling an odd number or a prime
number?
Again, A = {1, 3, 5} and B = {2, 3, 5} so
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
14. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
(b) What is the probability of rolling an odd number or a prime
number?
Again, A = {1, 3, 5} and B = {2, 3, 5} so
A ∪ B = {1, 2, 3, 5}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
15. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
(b) What is the probability of rolling an odd number or a prime
number?
Again, A = {1, 3, 5} and B = {2, 3, 5} so
A ∪ B = {1, 2, 3, 5}
By the equally likely assumption
n(A ∪ B) 4 2
P(A ∪ B) = = =
n(S) 6 3
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
16. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
17. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
18. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?
Only if A and B are mutually exclusive (disjoint), that is,
if A ∩ B = ∅.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
19. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?
Only if A and B are mutually exclusive (disjoint), that is,
if A ∩ B = ∅.
In this case, P(A ∪ B) is the sum of the probabilities of all
of the simple events in A plus the sum of the probabilities
of all of the simple events in B.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
20. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
But what happens if A and B are not mutually exclusive;
that is, what if A ∩ B = ∅?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
21. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
But what happens if A and B are not mutually exclusive;
that is, what if A ∩ B = ∅?
In this case, we must use a version of the addition principle
for probability.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
22. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Definition (Probability of a Union of Two Events)
For any events A and B,
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive, then
P(A ∪ B) = P(A) + P(B)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
23. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example Let E and F be events in a sample space S. If
P(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what is
P(E ∩ F )?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
24. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example Let E and F be events in a sample space S. If
P(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what is
P(E ∩ F )?
Notice here that P(E ∪ F ) = P(E) + P(F ), so we know that the
events E and F are not mutually exclusive; that is,
P(E ∩ F ) = 0.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
25. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example Let E and F be events in a sample space S. If
P(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what is
P(E ∩ F )?
Notice here that P(E ∪ F ) = P(E) + P(F ), so we know that the
events E and F are not mutually exclusive; that is,
P(E ∩ F ) = 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F ) − P(E ∩ F )
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
26. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example Let E and F be events in a sample space S. If
P(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what is
P(E ∩ F )?
Notice here that P(E ∪ F ) = P(E) + P(F ), so we know that the
events E and F are not mutually exclusive; that is,
P(E ∩ F ) = 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F ) − P(E ∩ F )
0.55 = 0.35 + 0.25 − P(E ∩ F )
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
27. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example Let E and F be events in a sample space S. If
P(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what is
P(E ∩ F )?
Notice here that P(E ∪ F ) = P(E) + P(F ), so we know that the
events E and F are not mutually exclusive; that is,
P(E ∩ F ) = 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F ) − P(E ∩ F )
0.55 = 0.35 + 0.25 − P(E ∩ F )
P(E ∩ F ) = 0.35 + 0.25 − 0.55 = 0.05
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
28. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find the
probability that the card is a jack or club.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
29. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find the
probability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are E
and F mutually exclusive?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
30. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find the
probability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are E
and F mutually exclusive? No because a card can be both a
jack and a club?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
31. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find the
probability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are E
and F mutually exclusive? No because a card can be both a
jack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likely
assumption,
n(E) 4
P(E) = = .
n(S) 52
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
32. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find the
probability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are E
and F mutually exclusive? No because a card can be both a
jack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likely
assumption,
n(E) 4
P(E) = = .
n(S) 52
Also, n(F ) = 13 - there are thirteen clubs; by the equally
likely assumption,
n(F ) 13
P(F ) = = .
n(S) 52 ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
33. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;
by the equally likely assumption,
1
P(E ∩ F ) = .
52
Now we can use the addition principle to get
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
34. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;
by the equally likely assumption,
1
P(E ∩ F ) = .
52
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F ) − P(E ∩ F )
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
35. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;
by the equally likely assumption,
1
P(E ∩ F ) = .
52
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F ) − P(E ∩ F )
4 13 1
P(E ∪ F ) = + −
52 52 52
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
36. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;
by the equally likely assumption,
1
P(E ∩ F ) = .
52
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F ) − P(E ∩ F )
4 13 1
P(E ∪ F ) = + −
52 52 52
16 4
= =
52 13
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
37. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
38. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
(a) Use the multiplication principle to calculate the total number
of outcomes in the sample space.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
39. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
(a) Use the multiplication principle to calculate the total number
of outcomes in the sample space.
n(S) = (2)(2)(2) = 8
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
40. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
(a) Use the multiplication principle to calculate the total number
of outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
41. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
(a) Use the multiplication principle to calculate the total number
of outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
42. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
(a) Use the multiplication principle to calculate the total number
of outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.
Let A be the event that exactly two tails are flipped.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
43. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.
(Tossing three coins is the same experiment as tossing one
coin three times.)
(a) Use the multiplication principle to calculate the total number
of outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.
Let A be the event that exactly two tails are flipped.
Let B be the event that exactly three tails are flipped.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
44. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutually
exclusive; that is,
E = A ∪ B and A ∩ B = ∅
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
45. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutually
exclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
46. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutually
exclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
n({HTT , THT , TTH}) 3
P(A) = =
n(S) 8
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
47. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutually
exclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
n({HTT , THT , TTH}) 3
P(A) = =
n(S) 8
n({TTT }) 1
P(B) = =
n(S) 8
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
48. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutually
exclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
n({HTT , THT , TTH}) 3
P(A) = =
n(S) 8
n({TTT }) 1
P(B) = =
n(S) 8
Therefore,
3 1 1
P(E) = P(A) + P(B) = + =
8 8 2
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
49. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1 , . . . , en }
into two subsets E and E such that
E ∩ E = ∅.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
50. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1 , . . . , en }
into two subsets E and E such that
E ∩ E = ∅.
That is, E and E are mutually exclusive, and
E ∪ E = S.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
51. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1 , . . . , en }
into two subsets E and E such that
E ∩ E = ∅.
That is, E and E are mutually exclusive, and
E ∪ E = S.
Then E is called the complement of E relative to S.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
52. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1 , . . . , en }
into two subsets E and E such that
E ∩ E = ∅.
That is, E and E are mutually exclusive, and
E ∪ E = S.
Then E is called the complement of E relative to S. The set
E contains all the elements of S that are not in E.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
53. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Furthermore,
P(S) = P(E ∪ E )
= P(E) + P(E ) = 1
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
54. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Furthermore,
P(S) = P(E ∪ E )
= P(E) + P(E ) = 1
Therefore,
P(E) = 1 − P(E ) P(E ) = 1 − P(E)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
55. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Furthermore,
P(S) = P(E ∪ E )
= P(E) + P(E ) = 1
Therefore,
P(E) = 1 − P(E ) P(E ) = 1 − P(E)
Many times it is easier to first compute the probability that and
event won’t occur, and then use that to find the probability that
the event will occur.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
56. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
57. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
58. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
59. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
60. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0 = 0.9
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
61. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0 = 0.9
(c) P(A )
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
62. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0 = 0.9
(c) P(A )
P(A ) = 1 − P(A) = 1 − 0.6 = 0.4
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
63. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0 = 0.9
(c) P(A )
P(A ) = 1 − P(A) = 1 − 0.6 = 0.4
(d) P(A ∩ B )
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
64. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If A and B are mutually exclusive events with
P(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0 = 0.9
(c) P(A )
P(A ) = 1 − P(A) = 1 − 0.6 = 0.4
(d) P(A ∩ B )
Since A ∩ B = ∅, A ⊆ B so A ∩ B = A and
P(A ∩ B ) = P(A) = 0.6
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
65. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: What is the probability that when two dice are
tossed, the number of points on each die will not be the same?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
66. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: What is the probability that when two dice are
tossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. For
example,
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
67. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: What is the probability that when two dice are
tossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. For
example,
E be the set of all rolls of two dice which do not result in
doubles. Mathematically we can represent this as
E = {(n, m)|1 ≤ n, m ≤ 6 and n = m}
We wish to find P(E).
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
68. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: What is the probability that when two dice are
tossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. For
example,
E be the set of all rolls of two dice which do not result in
doubles. Mathematically we can represent this as
E = {(n, m)|1 ≤ n, m ≤ 6 and n = m}
We wish to find P(E). Let S be be the sample space for this
experiment.
S = {(n, m)|1 ≤ n, m ≤ 6} and n(S) = 36 ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
69. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Here we have
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
70. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Here we have
E = {(n, m)|1 ≤ n, m ≤ 6 and n = m}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
71. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Here we have
E = {(n, m)|1 ≤ n, m ≤ 6 and n = m}
= {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
72. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Here we have
E = {(n, m)|1 ≤ n, m ≤ 6 and n = m}
= {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
Since
n(E ) 6 1
P(E ) = = =
n(S) 36 6
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
73. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Here we have
E = {(n, m)|1 ≤ n, m ≤ 6 and n = m}
= {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
Since
n(E ) 6 1
P(E ) = = =
n(S) 36 6
we have
1 5
P(E) = 1 − P(E ) = 1 − =
6 6
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
74. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability that
heads turn up at least once?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
75. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability that
heads turn up at least once?
Let E represent the event “heads turns up at least once” and let
S represent the sample space. Notice first that
S = {HTTHT , HHHTT , THHHT , HTHTH, TTTTT , HHHHT , . . .}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
76. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability that
heads turn up at least once?
Let E represent the event “heads turns up at least once” and let
S represent the sample space. Notice first that
S = {HTTHT , HHHTT , THHHT , HTHTH, TTTTT , HHHHT , . . .}
We assume the coin is fair so that we may also assume that all
of the outcomes in the sample space are equally likely. What is
n(S)?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
77. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability that
heads turn up at least once?
Let E represent the event “heads turns up at least once” and let
S represent the sample space. Notice first that
S = {HTTHT , HHHTT , THHHT , HTHTH, TTTTT , HHHHT , . . .}
We assume the coin is fair so that we may also assume that all
of the outcomes in the sample space are equally likely. What is
n(S)?
n(S) = (2)(2)(2)(2)(2) = 32
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
78. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability that
heads turn up at least once?
Let E represent the event “heads turns up at least once” and let
S represent the sample space. Notice first that
S = {HTTHT , HHHTT , THHHT , HTHTH, TTTTT , HHHHT , . . .}
We assume the coin is fair so that we may also assume that all
of the outcomes in the sample space are equally likely. What is
n(S)?
n(S) = (2)(2)(2)(2)(2) = 32
E contains all outcomes that have at least one H. E.g. HTTHT ,
HHHTT , etc.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
79. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
What is in the set E ?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
80. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
What is in the set E ? The opposite of “heads turn up at least
once” is “heads do not turn up at all.” So,
1
E = {TTTTT } and P(E ) =
32
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
81. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
What is in the set E ? The opposite of “heads turn up at least
once” is “heads do not turn up at all.” So,
1
E = {TTTTT } and P(E ) =
32
Therefore,
1 31
P(E) = 1 − P(E ) = 1 − =
32 32
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
82. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
What is in the set E ? The opposite of “heads turn up at least
once” is “heads do not turn up at all.” So,
1
E = {TTTTT } and P(E ) =
32
Therefore,
1 31
P(E) = 1 − P(E ) = 1 − =
32 32
Tip: Consider using complements whenever you encounter a
probability (or even counting problems) that contains the phrase
“at least once”.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
83. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A shipment of 40 precision parts, including 8 that are
defective, is sent to an assembly plant. The quality control
division selects 10 at random for testing and rejects the
shipment if 1 or more in the sample are found defective. What
is the probability that the shipment will be rejected?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
84. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: A shipment of 40 precision parts, including 8 that are
defective, is sent to an assembly plant. The quality control
division selects 10 at random for testing and rejects the
shipment if 1 or more in the sample are found defective. What
is the probability that the shipment will be rejected?
Notice first that the question
What is the probability that the shipment will be
rejected?
is really asking
What is the probability that the 10 parts selected for
testing contain at least one defective part?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
85. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
86. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
n(S) = C(40, 10) = 847, 660, 528
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
87. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
n(S) = C(40, 10) = 847, 660, 528
Let E be the set of all selections of 10 parts that contain at least
one defective part. We want to find P(E).
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
88. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
n(S) = C(40, 10) = 847, 660, 528
Let E be the set of all selections of 10 parts that contain at least
one defective part. We want to find P(E).
Notice that every set of 10 parts either
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
89. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
n(S) = C(40, 10) = 847, 660, 528
Let E be the set of all selections of 10 parts that contain at least
one defective part. We want to find P(E).
Notice that every set of 10 parts either
contains at least one defective part (so is in E), or
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
90. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
n(S) = C(40, 10) = 847, 660, 528
Let E be the set of all selections of 10 parts that contain at least
one defective part. We want to find P(E).
Notice that every set of 10 parts either
contains at least one defective part (so is in E), or
contains no defective parts
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
91. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of
40.
n(S) = C(40, 10) = 847, 660, 528
Let E be the set of all selections of 10 parts that contain at least
one defective part. We want to find P(E).
Notice that every set of 10 parts either
contains at least one defective part (so is in E), or
contains no defective parts
Thus E is the set of all selections of 10 parts that contain no
defective parts.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
92. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick
10 that have no defective parts, we choose from the 32 that are
not defective, so
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
93. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick
10 that have no defective parts, we choose from the 32 that are
not defective, so
n(E ) = C(32, 10) = 64, 512, 240
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
94. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick
10 that have no defective parts, we choose from the 32 that are
not defective, so
n(E ) = C(32, 10) = 64, 512, 240
Therefore,
n(E )
P(E) = 1 − P(E ) = 1 −
n(S)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
95. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick
10 that have no defective parts, we choose from the 32 that are
not defective, so
n(E ) = C(32, 10) = 64, 512, 240
Therefore,
n(E )
P(E) = 1 − P(E ) = 1 −
n(S)
64, 512, 240
=1− ≈ 1 − 0.0761
847, 660, 528
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
96. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick
10 that have no defective parts, we choose from the 32 that are
not defective, so
n(E ) = C(32, 10) = 64, 512, 240
Therefore,
n(E )
P(E) = 1 − P(E ) = 1 −
n(S)
64, 512, 240
=1− ≈ 1 − 0.0761
847, 660, 528
≈ 0.9239
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
97. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick
10 that have no defective parts, we choose from the 32 that are
not defective, so
n(E ) = C(32, 10) = 64, 512, 240
Therefore,
n(E )
P(E) = 1 − P(E ) = 1 −
n(S)
64, 512, 240
=1− ≈ 1 − 0.0761
847, 660, 528
≈ 0.9239
So there is about a 92.4% chance that the shipment will be
../images/stackedlogo-bw-
rejected.
Jason Aubrey Math 1300 Finite Mathematics
98. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
99. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E, then
1 the odds for E are given by
P(E) P(E)
= , P(E) = 1.
1 − P(E) P(E )
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
100. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E, then
1 the odds for E are given by
P(E) P(E)
= , P(E) = 1.
1 − P(E) P(E )
2 the odds against E are given by
1 − P(E) P(E )
= , P(E) = 0
P(E) P(E)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
101. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E, then
1 the odds for E are given by
P(E) P(E)
= , P(E) = 1.
1 − P(E) P(E )
2 the odds against E are given by
1 − P(E) P(E )
= , P(E) = 0
P(E) P(E)
Note: When possible, odds are to be expressed as ratios of
whole numbers. ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
102. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Given the following probabilities for an event E, find
the odds for and against E:
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
103. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Given the following probabilities for an event E, find
the odds for and against E:
3
P(E) = 5
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
104. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Given the following probabilities for an event E, find
the odds for and against E:
P(E) = 35
Since P(E) = 3/5 we know P(E ) = 1 − P(E) = 2/5.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
105. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Given the following probabilities for an event E, find
the odds for and against E:
P(E) = 35
Since P(E) = 3/5 we know P(E ) = 1 − P(E) = 2/5.
Then the odds for E are
P(E) 3/5 3
= =
P(E ) 2/5 2
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
106. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Given the following probabilities for an event E, find
the odds for and against E:
P(E) = 35
Since P(E) = 3/5 we know P(E ) = 1 − P(E) = 2/5.
Then the odds for E are
P(E) 3/5 3
= =
P(E ) 2/5 2
And the odds against E are
P(E ) 2/5 2
= =
P(E) 3/5 3
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
107. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
P(E) = 0.35
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
108. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
P(E) = 0.35
Since P(E) = 0.35 we know P(E ) = 1 − 0.35 = 0.65
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
109. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
P(E) = 0.35
Since P(E) = 0.35 we know P(E ) = 1 − 0.35 = 0.65
Then the odds for E are
P(E) 0.35 7
= =
P(E ) 0.65 13
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
110. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
P(E) = 0.35
Since P(E) = 0.35 we know P(E ) = 1 − 0.35 = 0.65
Then the odds for E are
P(E) 0.35 7
= =
P(E ) 0.65 13
And the odds against E are
P(E ) 0.65 13
= =
P(E) 0.35 7
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
111. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven when
two dice are tossed.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
112. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven when
two dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
113. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven when
two dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
n(E) = 6, n(S) = 36,
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
114. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven when
two dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
n(E) = 6, n(S) = 36,
6 30
P(E) = and P(E ) =
36 36
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
115. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven when
two dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
n(E) = 6, n(S) = 36,
6 30
P(E) = and P(E ) =
36 36
Therefore
P(E) 6/36 6 1
= = =
P(E ) 30/36 30 5
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
116. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
a
If the odds for an event E are , then the probability of E
b
is,
a
P(E) =
a+b
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
117. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
a
If the odds for an event E are , then the probability of E
b
is,
a
P(E) =
a+b
a
If the odds against an event E are then the probability of
b
E is
b
P(E) =
a+b
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
118. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds against
rolling a 6 before rolling a 7 are 6:5, what is the probability of
rolling a 6 before rolling a 7?
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
119. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds against
rolling a 6 before rolling a 7 are 6:5, what is the probability of
rolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
120. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds against
rolling a 6 before rolling a 7 are 6:5, what is the probability of
rolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.
odds against E are 6:5
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
121. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds against
rolling a 6 before rolling a 7 are 6:5, what is the probability of
rolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.
odds against E are 6:5
Therefore,
5 5
P(E) = =
6+5 11
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
122. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: The data below was obtained from a random survey
of 1,000 residents of a state. The participants were asked their
political affiliations and their preferences in an upcoming
gubernatorial election (D = Democrat, R = Republican, U =
Unaffiliated. )
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
123. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
Example: The data below was obtained from a random survey
of 1,000 residents of a state. The participants were asked their
political affiliations and their preferences in an upcoming
gubernatorial election (D = Democrat, R = Republican, U =
Unaffiliated. )
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
If a resident of the state is selected at random, what is the
empirical probability that the resident is not affiliated with a
political party or has no preference? What are the odds for this
../images/stackedlogo-bw-
event?
Jason Aubrey Math 1300 Finite Mathematics
124. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
We are looking for P(U ∪ N):
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
125. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N) − P(U ∩ N)
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
126. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N) − P(U ∩ N)
150 85 15
= + −
1000 1000 1000
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
127. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N) − P(U ∩ N)
150 85 15
= + −
1000 1000 1000
220
= = 0.22 or 22%
1000
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
128. Union and Intersection
Complement of an Event
Odds
Applications to Empirical Probability
D R U Totals
Candidate A 200 100 85 385
Candidate B 250 230 50 530
No Preference 50 20 15 85
Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N) − P(U ∩ N)
150 85 15
= + −
1000 1000 1000
220
= = 0.22 or 22%
1000
Then the odds for this event are
22 22 11
= =
100 − 22 78 39 ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics