Probability of Compound Events
Sample Space:
Let event A = "drawing a jack"
A=
{
,
,
,
P(A) =Probability of drawing a jack =
}
number of ways A can occur
= 4/52 = 1/13
number of possible outcomes
Let event B = "drawing a ten"
B ={
,
,
,
P(B) = Probability of drawing a ten =
}
number of ways B can occur
= 4/52 = 1/13
number of possible outcomes
The events A and B are said to be disjoint or mutually exclusive because
they do not anything in common.
A or B=
{
, , , ,
,
,
P(A or B) = Probability of drawing a jack or ten = 8/52 = 2/13
P(A) + P(B) = 1/13 + 1/13 = 2/13
In general, for disjoint events, P(A or B) = P(A) + P(B)
,
}
Let event A = "drawing a jack"
A=
{
,
,
}
,
P(A) = Probability of drawing a jack = 4/52 = 1/13
Let event C = "drawing a heart"
C=
{
,
,
,
,
,
,
,
,
,
,
,
,
}
P(C) = Probability of drawing a heart = 13/52 = 1/4
Events A and C are not disjoint because they have a jack of hearts in
common.
Event (A and C) =
{ }
P(A and C) = 1/52
Event (A or C) =
{
,
,
,
,
={
,
,
,
,
} or {
}
,
,
,
,
,
,
}
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
P(A or C) = 16/52 = 4/13
In general, P(A or C) = P(A) + P(C) - P(A and C)
= 1/13 + 1/4 - 1/52 = 4/13
In general, for non-disjoint events, P(A or B) = P(A) + P(B) - P(A and B)
Let event D = "drawing a face card"
D=
{
,
,
,
,
,
,
,
,
}
,
,
,
Let even E = "drawing a spade"
E=
{
,
,
,
D and E =
,
,
,
,
}
{
,
,
}
,
,
,
,
,
P(D) = Probability of drawing a face card = 12/52 = 3/13
P(E) = Probability of drawing a spade = 13/52 = 1/4
P(D and E) = Probability of drawing a face card that is a spade = 3/52
P(D or E) = Probability of drawing a face card or a spade =
= P(D) + P(E) - P(D and E) = 3/13 + 1/4 - 3/52 = 11/26
D or E =
{
,
,
,
,
,
,
,
,
,
,
}
,
,
,
or
{
,
,
,
,
,
,
={
,
,
,
,
,
,
,
,
,
,
,
P(D or E) = 22/52 = 11/26
,
,
,
,
,
}
,
,
,
,
,
,
}
,
,
,
Complement of an Event
Let event A = "drawing a 2"
A=
{
,
,
,
}
Then the complement of event A = "drawing a card that is not a 2"
Complement of A
={
}
There are four cards with a 2.
Thus, P(A) = 4/52 = 1/13
There are 48 cards that are not a 2.
Thus, P(complement of A) = 48/52 = 12/13
We can see that P(A) + P(complement of A) = 1/13 + 12/13 = 13/13 = 1
Or P(A) = 1 - P(complement of A)
In general,
P(A) + P(Complement of A) = 1; or
P(A) = 1 - P(Complement of A)
Let event C = "drawing a club"
C={
,
,
,
,
,
,
,
,
,
,
,
,
}
Then the complement of event C = "drawing a card that is not a spade"
Complement of C =
{
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
}
There are thirteen spades.
Thus, P(C) = 13/52 = 1/4
There are 39 cards that are not spade.
Thus, P(complement of C) = 39/52 = 3/4
P(C) + P(complement of C) = 1/4 + 3/4 = 1
Or P(C) = 1 - P(complement of C)
The following table shows how people in the US identify
themselves on economic and social values.
Conservative
Moderate
Liberal
Other
Economic issues
46%
32%
20%
2%
Social issues
38%
31%
28%
3%
Source: Gallup's annual Values and Beliefs poll, conducted May 3-6, 2012.
If a person is selected randomly, what is the probability that the
person is a not a liberal on social issues?
Answer:
Complement of " not a liberal on social issues" is "liberal on social issues"
P(not a liberal on social issues) = 1 - P(liberal on social issues) = 1 - 20% =
100% - 20% = 80%
Presidential General Election Results (National 2008)
Party
Percentage
Democratic
Republican
Independent
Other
52.87%
45.60%
0.00%
1.53%
Source: http://uselectionatlas.org
If a person is selected randomly, what is the probability that the
person did not vote for a Republican candidate?
Answer:
Complement of "did not vote for a Republican candidate" is "did vote for
a Republican candidate"
P(did not vote for a Republican candidate)
= 1 - P(did vote for a Republican candidate)
= 1 - 45.60% = 100% - 45.60% = 54.5%
The following data shows religious preference of Americans.
These results are based on a compilation of 327,244 interviews
conducted as part of Gallup Daily tracking from January-November 2011.
Religious Preference
Protestant/Other Christian
Catholic
Mormon
Jewish
Muslim
Other non-Christian religion
None/Atheist/Agnostic
No response given
Percentage
52.5%
23.6%
1.9%
1.6%
0.5%
2.4%
15.0%
2.5%
Source: http://www.gallup.com/poll/151760/Christianity-Remains-Dominant-Religion-United-States.aspx
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Preference for Presidential Nominee
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
Total
6817 33463
47721
88001
If one voter is selected randomly, what is the probability that the voter
voted for Ron Paul?
Answer: 10,621/88001 = 12.07%
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Preference for Presidential Nominee
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
Total
6817 33463
47721
88001
If an "Early" voter is selected randomly, what is the probability that the
voter voted for Mitt Romney?
Answer: 22,797/33463 = 33463
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Preference for Presidential Nominee
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
Total
6817 33463
47721
88001
If an "Election" voter is selected randomly, what is the probability that
the voter voted for Rick Santorum?
Answer: 5,562/47721 = 11.66%
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
Total
6817 33463
47721
88001
If an "Early" voter is selected randomly, what is the probability that the
voter voted for Mitt Romney?
Answer = 22,797/33463 = 68.13%
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Preference for Presidential Nominee
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Total
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
6817 33463
47721
88001
If an "Election" voter is selected randomly, what is the probability that
the person voted for Mitt Romney or John Huntsman?
P("Election" voter voted for John Huntsman) = 304/47721 = 0.00637
P("Election" voter voted for Mitt Romney) = 30,935/47721 = 0.6482
P("Election" voter voted for John Huntsman AND Mitt Romney) =
0/47721 = 0
P("Election" voter voted for John Huntsman OR Mitt Romney) =
P("Election" voter voted for John Huntsman)
+P("Election" voter voted for Mitt Romney)
- P("Election" voter voted for John Huntsman AND Mitt Romney)
= 0.00637 + 0.6428 - 0 = 0.64917
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Preference for Presidential Nominee
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Total
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
6817 33463
47721
88001
If a person voted for John Davis, what is the probability that the voter
voted on election day?
Answer = 148/418 = 35.4%
Republican Party Cumulative Report — Unofficial
Tarrant County — Primary Election — May 29, 2012
Preference for Presidential Nominee
Candidate
John Davis
Michele Bachmann
Charles "Buddy" Roemer
Newt Gingrich
Ron Paul
Jon Huntsman
Mitt Romney
Rick Santorum
Uncommitted
Absentee Early Election Total
11
259
148
418
39
320
408
767
10
75
98
183
286 1,451
2,181
3,918
440 3,639
6,542 10,621
22
249
304
575
5,344 22,797
30,935 59,076
417 3,676
5,562
9,655
248
997
1,543
2,788
Total
6817 33463
47721
88001
If a person voted for Michele Bachmann is selected randomly, what is the
probability
that the person did early voting?
Answer = 320/767 = 41.72%