event - any collection of results or outcomes from some procedure
DESCRIPTION
Event - any collection of results or outcomes from some procedure Simple event - any outcome or event that cannot be broken down into simpler components - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/1.jpg)
1
Event - any collection of results or outcomes from some procedure
Simple event - any outcome or event that cannot be broken down into simpler components
Compound event – an event made up of two or more other events
Sample space - all possible simple events
Definitions
![Page 2: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/2.jpg)
2
Notation P - denotes a probability
A, B, ... - denote specific events
P (E) - denotes the probability of event E occurring
![Page 3: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/3.jpg)
3
Basic Rules for Computing Probability
Rule 1: Relative Frequency Approximation
Conduct (or observe) an experiment a large number of times, and count the number of times event E actually occurs, then an estimate of P(E) is
![Page 4: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/4.jpg)
4
Basic Rules for Computing Probability
Rule 1: Relative Frequency Approximation
Conduct (or observe) an experiment a large number of times, and count the number of times event A actually occurs, then an estimate of P(E) is
P(E) = Number of times E occurredTotal number of possible outcomes
![Page 5: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/5.jpg)
5
Basic Rules for Computing Probability
Rule 2: Classical approach (requires equally likely outcomes)
If a procedure has n different simple events, each with an equal chance of occurring, and s is the number of ways event E can occur, then
![Page 6: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/6.jpg)
6
Basic Rules for Computing Probability
Rule 2: Classical approach (requires equally likely outcomes)
If a procedure has n different simple events, each with an equal chance of occurring, and s is the number of ways event E can occur, then
P(E) = number of ways E can occurnumber of different simple
events
sn =
![Page 7: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/7.jpg)
7
Basic Rules for Computing Probability
Rule 3: Subjective Probabilities
P(E), the probability of E, is found by simply guessing or
estimating its value based on knowledge of the relevant
circumstances.
![Page 8: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/8.jpg)
8
Rule 1 The relative frequency approach is an
approximation.
![Page 9: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/9.jpg)
9
Rule 1 The relative frequency approach is an
approximation.
Rule 2 The classical approach is the actual
probability.
![Page 10: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/10.jpg)
10
Law of Large Numbers
As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability.
![Page 11: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/11.jpg)
11
Law of Large Numbers
Flip a coin 20 times and record the number of heads after each trial. In L1 list the numbers 1-20, in L2 record the number of heads.In L3, divide L2 by L1. Get a scatter plot with L1 and L3. What can you conclude?
![Page 12: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/12.jpg)
12
The sample space consists of two simple events: the person is struck by lightning or is not. Because these simple events are not equally likely, we can use the relative frequency approximation (Rule 1) or subjectively estimate the probability (Rule 3). Using Rule 1, we can research past events to determine that in a recent year 377 people were struck by lightning in the US, which has a population of about 274,037,295. Therefore,
P(struck by lightning in a year)
377 / 274,037,295 1/727,000
Example: Find the probability that a randomly selected person will be struck by lightning this year.
![Page 13: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/13.jpg)
13
Example: On an ACT or SAT test, a typical multiple-choice question has 5 possible answers. If you make a random guess
on one such question, what is the probability that your response is wrong?
There are 5 possible outcomes or answers, and there are 4 ways to answer incorrectly. Random guessing implies that the outcomes in the sample space are equally likely, so we apply the classical approach (Rule 2) to get:
P(wrong answer) = 4 / 5 = 0.8
![Page 14: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/14.jpg)
14
Probability Limits
The probability of an impossible event is 0.
The probability of an event that is certain to occur is 1.
![Page 15: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/15.jpg)
15
Probability Limits
The probability of an impossible event is 0.
The probability of an event that is certain to occur is 1.
• 0 P(A) 1
![Page 16: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/16.jpg)
16
Probability Limits
The probability of an impossible event is 0.
The probability of an event that is certain to occur is 1.
0 P(A) 1
Impossibleto occur
Certainto occur
![Page 17: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/17.jpg)
17
Possible Values for ProbabilitiesCertain
Likely
50-50 Chance
Unlikely
Impossible
1
0.5
0
![Page 18: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/18.jpg)
18
Complementary Events
![Page 19: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/19.jpg)
19
Complementary Events
The complement of event E, denoted by Ec, consists of all outcomes in which event E
does not occur.
![Page 20: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/20.jpg)
20
P(E)
Complementary Events
The complement of event E, denoted by Ec, consists of all outcomes in which event E does not occur.
P(EC)(read “not E”)
![Page 21: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/21.jpg)
21
Example: Testing CorvettesThe General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver?
![Page 22: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/22.jpg)
22
Because 20 of the 50 subjects are men, it follows that 30 of the 50 subjects are women so,
P(not selecting a man) = P(man)c
= P(woman) = 30 = 0.6
50
Example: Testing CorvettesThe General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver?
![Page 23: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/23.jpg)
23
Using a Tree Diagram
Flipping a coin is an experiment and the possible outcomes are heads (H) or tails (T).
One way to picture the outcomes of an experiment is to draw a tree diagram. Each outcome is shown on a separate branch. For example, the outcomes of flipping a coin are
H
T
![Page 24: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/24.jpg)
24
A Tree Diagram for Tossing a Coin Twice
There are 4 possible outcomes when tossing a coin twice.
H
TH
T
H
T
First Toss Second Toss Outcomes
HH
HTTH
TT
![Page 25: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/25.jpg)
25
Rules of Complementary Events
P(A) + P(A)c = 1
![Page 26: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/26.jpg)
26
P(A)c
Rules of Complementary Events
P(A) + P(A)c = 1
= 1 - P(A)
![Page 27: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/27.jpg)
27
(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)
Possible outcomes for two rolls of a die
![Page 28: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/28.jpg)
28
1. Find the probability that the sum is a 22. Find the probability that the sum is a 33. Find the probability that the sum is a 44. Find the probability that the sum is a 55. Find the probability that the sum is a 66. Find the probability that the sum is a 77. Find the probability that the sum is a 88. Find the probability that the sum is a 99. Find the probability that the sum is a 1010. Find the probability that the sum is a 1111. Find the probability that the sum is a 12
• 1/36• 2/36• 3/36• 4/36• 5/36• 6/36• 5/36• 4/26• 3/36• 2/36• 1/36
Find the following probabilities
![Page 29: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/29.jpg)
29
![Page 30: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/30.jpg)
30
Rounding Off Probabilities
give the exact fraction or decimal
or
![Page 31: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/31.jpg)
31
Rounding Off Probabilities
give the exact fraction or decimal
orround off the final result to three significant digits
![Page 32: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/32.jpg)
32
How many ways are there to answer a two question test when the first question is a true-false question and the second question is a multiple choice question with five possible answers?
Tree Diagram of Test Answers
![Page 33: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/33.jpg)
33
TaTbTcTdTeFaFbFcFdFe
a
b
c
d
e
a
b
c
d
e
T
F
Tree Diagram of Test Answers
![Page 34: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/34.jpg)
34
What is the probability that the first question is true and the second
question is c?
Tree Diagram of Test Answers
![Page 35: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/35.jpg)
35
TaTbTcTdTeFaFbFcFdFe
a
b
c
d
e
a
b
c
d
e
T
F
Tree Diagram of Test Answers
![Page 36: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/36.jpg)
36
TaTbTcTdTeFaFbFcFdFe
a
b
c
d
e
a
b
c
d
e
T
F
P(T) =
FIGURE 3-9 Tree Diagram of Test Answers
12
![Page 37: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/37.jpg)
37
TaTbTcTdTeFaFbFcFdFe
a
b
c
d
e
a
b
c
d
e
T
F
P(T) = P(c) =
Tree Diagram of Test Answers
12
15
![Page 38: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/38.jpg)
38
TaTbTcTdTeFaFbFcFdFe
a
b
c
d
e
a
b
c
d
e
T
F
P(T) = P(c) = P(T and c) =
FIGURE 3-9 Tree Diagram of Test Answers
12
15
110
![Page 39: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/39.jpg)
39
P (both correct) = P (T and c)
5
1
2
1
10
1
![Page 40: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/40.jpg)
40
R P S
R P S R P S R P S
Rock – Paper – Scissors Tree Diagram- 2 Players
T B A A T B B A T
3
1
9
3)(
3
1
9
3)(
3
1
9
3)( TPBPAP
![Page 41: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/41.jpg)
41
R P S
R P S R P S R P S
RPSRP SRP SRPSRPSRPSRP SRPSRPS
Rock – Paper – Scissors Tree Diagram- 3 Players
A B B B B C B C B B B C B A B C B B B C B C B B B B A
9
2
27
6)(
3
2
27
18)(
9
1
27
3)( CPBPAP
![Page 42: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/42.jpg)
42
1/165/8
13/14
1/3
3/4
1/3
Pg 189 #9
![Page 43: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/43.jpg)
43
Compound Event• Any event combining 2 or more simple events
Definition
![Page 44: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/44.jpg)
44
![Page 45: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/45.jpg)
45
Notation
• P(A or B) = P (event A occurs or event B occurs or they both occur)
Definition
![Page 46: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/46.jpg)
46
• General Rule
When finding the probability that event A occurs or
event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find the total in such a way that no outcome is counted more than once.
Compound Event
![Page 47: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/47.jpg)
47
• Formal Addition Rule• P(A or B) = P(A) + P(B) - P(A and B)
• where P(A and B) denotes the probability that A and B both occur at the same time.
Compound Event
B)P(AP(B)P(A)B)P(A
![Page 48: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/48.jpg)
48
Definition• Events A and B are mutually exclusive if
they cannot occur simultaneously.
![Page 49: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/49.jpg)
49
Venn Diagrams
Total Area = 1
P(A) P(B)
Non-overlapping Events
P(A) P(B)
P(A and B)
Overlapping Events
Total Area = 1
![Page 50: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/50.jpg)
50Venn Diagrams
![Page 51: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/51.jpg)
51
A (B C) Ac B (A B )c
A (B C)(A B) C A - B
(AB) (A C) U - Ac Ac Bc Cc
![Page 52: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/52.jpg)
52
Ac (Bc C)Ac (B C)c (A B)c
(A B) (A C) A - (B C) A (B Ac)
(A B) (A C) B - A (Ac Bc) Cc
![Page 53: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/53.jpg)
53
A poll was taken of 100 students to find out how they arrived at school. 28 used car pools; 31 used buses; and 42 said they drove to school alone.In addition, 9 used both car pools and buses;10 used car pools and drove alone; only 6 used buses and their own car and 4 used all three methods.
a. Complete the Venn diagram.b. How many used none of the methods?c. How many used only car pools?d. How many used buses exclusively?
A B
UC
46 5
230 20
1320
201320
P(ABC) P(A) P(B) P(C) P(AB) P(BC) P(AC) P(ABC)
P(A U B U C) = 42 + 31 + 28 – 6 – 10 -9 + 4 = 80
![Page 54: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/54.jpg)
54
A survey of 500 television watchers produced the following information:285 watch football190 watch hockey115 watch basketball45 watch football and basketball70 watch football and hockey50 watch hockey and basketball50 do not watch any sports.
a. How many watch all three games?b. How many watch exactly one of the three games?
A B
UC
P(ABC) P(A) P(B) P(C) P(AB) P(BC) P(AC) P(ABC)
500 = 285 + 190 + 115 - 45 - 70 - 50 + P(A B C) + 50
500 = 475 + P(A B C) P(A B C) = 25
2545 20
45195
95
25
50
![Page 55: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/55.jpg)
55
Applying the Addition Rule
P(A or B)
Addition Rule
AreA and Bmutuallyexclusive
?
P(A or B) = P(A)+ P(B) - P(A and B)
P(A or B) = P(A) + P(B)Yes
No
![Page 56: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/56.jpg)
56
• Find the probability of randomly selecting a man or a boy.
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
![Page 57: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/57.jpg)
57
• Find the probability of randomly selecting a man or a boy.
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
![Page 58: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/58.jpg)
58
• Find the probability of randomly selecting a man or a boy.
• P(man or boy) =
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
.7902223
1756
2223
64
2223
1692
* Mutually Exclusive *
![Page 59: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/59.jpg)
59
• Find the probability of randomly selecting a man or someone who survived.
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
![Page 60: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/60.jpg)
60
• Find the probability of randomly selecting a man or someone who survived.
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
![Page 61: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/61.jpg)
61
• Find the probability of randomly selecting a man or someone who survived.
• P(man or survivor) =
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
* NOT Mutually Exclusive *
= 0.9292223
2066
2223
332
2223
706
2223
1692
![Page 62: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/62.jpg)
62
Complementary Events
P(A) and P(A)c are mutually exclusive
All simple events are either in A or Ac
![Page 63: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/63.jpg)
63
Venn Diagram for the Complement of Event A
Total Area = 1
P (A)
P (A)c = 1 - P (A)
![Page 64: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/64.jpg)
64
Finding the Probability of Two or More Selections
Multiple selections
Multiplication Rule
![Page 65: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/65.jpg)
65
NotationP(A and B) =
P(event A occurs in a first trial and
event B occurs in a second trial)
![Page 66: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/66.jpg)
66
Conditional Probability
Definition
The conditional probability of event B occurring, given that A has already occurred, can be found by dividing the probability of events A and B both occurring by the probability of event A.
![Page 67: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/67.jpg)
67
Conditional Probability
P(A and B) = P(A) • P(B|A)
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)
B)P(A
P(A and B)P(A)P(B|A) =
![Page 68: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/68.jpg)
68
P(B|A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B|A as “B given A”).
Notation for Conditional Probability
![Page 69: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/69.jpg)
69
Definitions
Independent 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.
![Page 70: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/70.jpg)
70
Formal Multiplication Rule
P(A and B) = P(A) • P(B|A)
If A and B are independent events, P(B|A) is really the same as P(B)
![Page 71: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/71.jpg)
71
Applying the Multiplication Rule
P(A or B)
Multiplication Rule
AreA and B
independent?
P(A and B) = P(A) • P(B|A)
P(A and B) = P(A) • P(B)Yes
No
![Page 72: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/72.jpg)
72
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)c
where P(A)c is P(getting none)
![Page 73: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/73.jpg)
73
Probability of ‘At Least One’ Find the probability of a couple have
at least 1 girl among 3 children. If P(A) = P(getting at least 1 girl), then
P(A) = 1 - P(A)c
where P(A)c is P(getting no girls)
P(A)c = (0.5)(0.5)(0.5) = 0.125
P(A) = 1 - 0.125 = 0.875
![Page 74: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/74.jpg)
74
If P(B|A) = P(B)
then the occurrence of A has no effect on the probability of event B; that is, A and B are independent events.
Testing for Independence
![Page 75: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/75.jpg)
75
If P(B|A) = P(B)
then the occurrence of A has no effect on the probability of event B; that is, A and B are independent events.
or
If P(A and B) = P(A) • P(B)
then A and B are independent events.
Testing for Independence
![Page 76: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/76.jpg)
76
Find the probability of randomly selecting a man if you know the person is a survivor
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table
)P(survivor
survivor)P(mansurvivor)|P(man
.470
706
332
)P(b
survivor)P(bboy)anot |P(survivor
c
c .309
2159
677
Find the probability of selecting a survivor if you know the person is not a boy
![Page 77: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/77.jpg)
77
Calculate the following probabilities:
a. P(A1) e. P(A2 U B3) b. P(B3) f. P(B1 U B4)c. P (A1 B4) g. P( B2 B4)d. P(B1|A3) h. P(A2|B4)
East B1 South B2 Midwest B3 Farwest B4
City Type
Large A1 35 10 25 25
Small A2 15 25 15 15
Suburb A3 25 5 10 10
75 40 50 50
957050
215
95/215= 19/43
50/215=10/43
25/215 = 5/43
25/50=1/2
105/215=21/43
125/215=25/43
none
15/50 = 3/10
![Page 78: Event - any collection of results or outcomes from some procedure](https://reader035.vdocuments.mx/reader035/viewer/2022081603/568138e9550346895da09cef/html5/thumbnails/78.jpg)
78
P(A) = 1/3P(B) = 1/4P(A U B) = 1/2
Find :
a. P(AB)
b. P(A | B)
c. P(B | A)
d. P(ABc)
e. P(Ac Bc)
f. P(Ac | B)
g. P(Ac | Bc)
h. P(Ac Bc)
1/123/12 2/12
6/12
A B
a. 1/12
b. 1/3
c. 1/4
d. 1/4
e. 1/2
f. 2/3
g. 2/3
h. 11/12