The probability of an event A is a measure of the likelihood of the event. Probabilities have the fo

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The probability of an event A is a measure of the
likelihood of the event. Probabilities have the
following properties
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If A1, A2, A3, , is a sequence of mutually
exclusive events, then
P(A1 or A2 or A3 or ) P(A1) P(A2) P(A3)
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It is certain (probability 1) that any event
either will or wont happen.
P(A U not(A)) P(A) P(not(A))
(complements are mutually exclusive)
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P(not(A)) 1 P(A)
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Example Suppose a fair 6-sided die is rolled.
What is the probability that an even number will
result?
S 1, 2, 3, 4, 5, 6 A 2, 4, 6
x P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6
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If an experiment can result in any one of N
different equally likely outcomes, and if exactly
n of these outcomes correspond to event A, then
the probability of event A is
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Example If 5 cards are dealt from a standard
deck of cards, what is the probability of have
all five cards being of the same suit?
A Five cards of the same suit S All five card
hands Operation 1 Choose a suit (n1) Operation
2 5 cards of same suit (n2)
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If A and B are two events, then
P(A or B) P(A) P(B) P(A and B)
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P(A or B or C) P(A) P(B) P(C) P(A and B)
P(A and C) P(B and C) P(A and B and C)
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
P(A) 0.07 0.09 0.13 0.15 0.44
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
P(B) 0.09 0.08 0.15 0.14 0.46
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
P(A B) 0.09 0.15 0.24
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
P(A or B) 0.070.090.080.130.150.14
0.66
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
P(not(B)) 0.07 0.13 0.11 0.23
0.54
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The conditional probability of B given A, denoted
by P(B A) is defined by
provided that P(A) gt 0
Note If P(A) 0 then event A is impossible
therefore it cannot be given to have happened.
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Venn Diagram Population Obstructive Lung Disease
A Chronic bronchitis B Emphysema C Asthma
P(AB) P(A B) / P(B) (0.090.15)/(0.090.15
0.080.14) 0.52174
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If in an experiment events A and B can occur, then
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Two events are independent if and only if
It is always true that P(A B)
P(A)P(BA) Under the special condition of
independence, P(A B) P(A)P(B) therefore, two
events are A and B independent if P(BA)
P(B) Interpretation of independence A and B are
independent if knowing whether A occurred or not
does not change the likelihood (probability) of B
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A B are mutually exclusive since P(A B) 0
Are A B independent? In order to be
independent P(B) P(BA) P(B) p2 P(BA) P(A
B)/P(A) 0/p1 0 Since p2 gt 0 then A and B
cannot be independent.
It is IMPOSSIBLE for events to be both mutually
exclusive and independent.
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An electrical system consists of four independent
components shown below along with the probability
that each component will operate properly. What
is the probability that the system will work?
0.95 (0.98) (0.75 0.70 0.75(0.70))
0.861175
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If the system does not work, what is the
probability that component B is not working?
Let Z system broken
Ways for system to not work when B component is
also not working
ABCD 0.95(0.02)(0.75)(0.70) 0.009975 ABCD
0.05(0.02)(0.75)(0.70) 0.000525 ABCD
0.95(0.02)(0.25)(0.70) 0.003325 ABCD
0.05(0.02)(0.25)(0.70) 0.000175 ABCD
0.95(0.02)(0.75)(0.30) 0.004275 ABCD
0.05(0.02)(0.75)(0.30) 0.000225 ABCD
0.95(0.02)(0.25)(0.30) 0.001425 ABCD
0.05(0.02)(0.25)(0.30) 0.000075
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If the system is working then what is the
probability that component C is also working?
Let Z system works
Calculating the probability of event (C and
Z) ABCD P(A and B and C and D)
0.95(0.98)(0.75)(0.70) 0.488775 ABCD P(A and B
and C and not(D)) 0.95(0.98)(0.75)(1-0.70)
0.209475 P(C and Z) 0.69825
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Homework 3 Pages 73 74 106, 112, 116, 120