GENERAL PROBABILITY RULES

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SECTION 6.3

• GENERAL PROBABILITY RULES

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Review of Previous Rules

• General addition rule
• P(A or B) P(A) P(B) P(A and B)
• Addition rule for disjoint events
• P(one or more of A, B, C) P(A) P(B) P(C)
• Multiplication rule for independent events
• P(A and B) P(A)P(B)

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Conditional Probability

• Conditional probability the probability of one
  event under the condition that we know another
  event.
• The can be interpreted as given the
  information that

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• General addition rule
• P(A or B) P(A) P(B) P(A and B)
• P(A U B) P(A) P(B) P(A n B)
• General multiplication rule
• P(A and B) P(A)P(B given A)
• P(A n B) P(A)P(BA)
• When P(A) gt 0,
• Testing for independence
• Two events A and B are independent if
• P(BA) P(B)

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TESTING FOR INDEPENDENCE

• Two events A and B are independent if
• P(BA) P(B)
• Think back to your last quiz. When rolling a die
  and then flipping a coin, let event A be getting
  a 1 or 2 on the roll of the die. Let event B be
  getting an even number on the die. Are A and B
  independent?

P(BA) (2/12)/(4/12) ½ P(B) 6/12 ½
Therefore, A and B are independent.
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A
BLUE REPRESENTS DESIGNATED AREA
A
B
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AC
BLUE REPRESENTS DESIGNATED AREA
A
B
8
B
BLUE REPRESENTS DESIGNATED AREA
A
B
9
BC
BLUE REPRESENTS DESIGNATED AREA
A
B
10
AnB
BLUE REPRESENTS DESIGNATED AREA
A
B
11
(AnB)C
BLUE REPRESENTS DESIGNATED AREA
A
B
12
AUB
BLUE REPRESENTS DESIGNATED AREA
A
B
13
(AUB)C
BLUE REPRESENTS DESIGNATED AREA
A
B
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Taste In Music

• Musical styles other than rock and pop are
  becoming more popular. A survey of college
  students find that 40 like country music, 30
  like gospel music, and 10 like both.
• What is the conditional probability that a
  student likes gospel music if we know that he/she
  likes country music?

Conditional Probability P(GC) 0.1/0.40.25
C
G
10
30
20
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Taste In Music (cont.)

• Musical styles other than rock and pop are
  becoming more popular. A survey of college
  students find that 40 like country music, 30
  like gospel music, and 10 like both.
• What is the conditional probability that a
  student who does not like country music likes
  gospel music?

Conditional Probability P(GCC) 0.2/0.61/3
C
G
10
30
20
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Venn Diagram PracticeRESTAURANT
T
C
15
12
9
11
7
13
10
23
Q
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RESTAURANT

• 1. (CUTUQ)C P(CUTUQ)C 23/100
• 2. (CnTnQ) P (CnTnQ) 11/100
• 3. (CUQnTC) P(CUQnTC) 29/100
• 4. (Q) P(Q) 41/100
• 5. (TnQ)U(QnC)U(CnT)
• P(TnQ)U(QnC)U(CnT) 46/100
• 6. (TnQnCC) P(TnQnCC) 13/100
• 7. (TnC) P (TnC) 26/100

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Venn Diagram PracticeCARTOONS
T
A
23
17
73
9
14
31
11
22
P
19
Venn Diagram PracticeCONCERT
D
D
P
11
27
21
10
13
15
35
18
G
G
20
Venn Diagram PracticeSTAR TREK
T
D
23
17
73
9
31
14
11
22
V
21
Venn Diagram PracticeMYTHOLOGY
L
H
5
12
18
3
6
7
24
25
R
22
Venn Diagram PracticePOLLUTANTS
P
C
137
122
101
28
72
152
211
177
S
23
Venn Diagram PracticeTENNIS
S
B
40
52
35
30
10
8
5
20
F
24
Venn Diagram PracticeTENNIS TOURNAMENTS
U
W
30
40
50
10
15
20
30
5
A
25
Venn Diagram PracticeLANGUAGES
F
G
8
29
20
4
16
12
92
27
S
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Nobel Prize Winners
COUNTRY PHYSICS CHEMISTRY Phys/Med Total
United States 74 51 90 215
United Kingdom 21 27 28 76
Germany 19 29 15 63
France 11 7 7 25
Soviet Union 9 1 2 12
Japan 4 4 0 8
TOTAL 138 119 142 399

• If a laureate is selected at random, what is the
  probability that
• his or her award is in chemistry?
• the award was won by someone working in the US?
• the awardee was working in the US, given the
  award was for phys./med?
• the award was for phys./med., given that the
  awardee was working in the US?

119/399 0.2982
215/399 0.5388
90/142 0.6338
90/215 0.4186