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SECTION 6.3. GENERAL PROBABILITY 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). Review of Previous Rules. - PowerPoint PPT Presentation
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SECTION 6.3
GENERAL PROBABILITY RULES
General addition ruleP(A or B) = P(A) + P(B) – P(A and B)
Addition rule for disjoint eventsP(one or more of A, B, C) = P(A) + P(B) + P(C)
Multiplication rule for independent eventsP(A and B) = P(A)P(B)
Review of Previous Rules
Conditional probability – the probability of one event under the condition that we know another event.
The “|” can be interpreted as “given the information that”
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 ∩ B)
General multiplication rule P(A and B) = P(A)P(B given A) P(A ∩ B) = P(A)P(B|A)
When P(A) > 0, Testing for independence:
Two events A and B are independent if P(B|A) = P(B)
TESTING FOR INDEPENDENCE
Two events A and B are independent if P(B|A) = 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(B|A) = (2/12)/(4/12) = ½
P(B) = 6/12 = ½
Therefore, A and B are independent.
A
A B
BLUE REPRESENTS DESIGNATED AREA
AC
A B
BLUE REPRESENTS DESIGNATED AREA
B
A B
BLUE REPRESENTS DESIGNATED AREA
BC
A B
BLUE REPRESENTS DESIGNATED AREA
A∩B
A B
BLUE REPRESENTS DESIGNATED AREA
(A∩B)C
A B
BLUE REPRESENTS DESIGNATED AREA
AUB
A B
BLUE REPRESENTS DESIGNATED AREA
(AUB)C
A B
BLUE REPRESENTS DESIGNATED AREA
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?
C G
10%30% 20%
Conditional Probability
P(G|C) = 0.1/0.4=0.25
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?
C G
10%30% 20%
Conditional Probability
P(G|CC) = 0.2/0.6=1/3
Venn Diagram PracticeRESTAURANT
11
23
13
15
7
10
129
T C
Q
RESTAURANT
1. (CUTUQ)C P(CUTUQ)C = 23/100 2. (C∩T∩Q) P (C∩T∩Q) = 11/100 3. (CUQ∩TC) P(CUQ∩TC) = 29/100 4. (Q) P(Q) = 41/100 5. (T∩Q)U(Q∩C)U(C∩T)
P(T∩Q)U(Q∩C)U(C∩T) = 46/100 6. (T∩Q∩CC) P(T∩Q∩CC) =
13/100 7. (T∩C) P (T∩C) = 26/100
Venn Diagram PracticeCARTOONS
T A
P 22
73
14319
23 17
11
Venn Diagram PracticeCONCERT
P D
G
D
G 18
1013
11
15
35
2721
Venn Diagram PracticeSTAR TREK
T D
V
73
31
22
14
1723
9
11
Venn Diagram PracticeMYTHOLOGY
L H
R25
37
5
6
24
1812
Venn Diagram PracticePOLLUTANTS
C P
S177
101
72
137
15228
122
211
Venn Diagram PracticeTENNIS
S B
F20
3010
40
5
5235
8
Venn Diagram PracticeTENNIS TOURNAMENTS
U W
A
30
10
1520
30
5
5040
Venn Diagram PracticeLANGUAGES
F G
S
416
820
27
12
92
29
Nobel Prize WinnersCOUNTRY PHYSIC
SCHEMISTR
YPhys/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 399If a laureate is selected at random, what is the probability that:
a) his or her award is in chemistry?
b) the award was won by someone working in the US?
c) the awardee was working in the US, given the award was for phys./med?
d) 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