j W f(x) = (x² /9 20 1px (a) The failure time in years (x) of a component has the probability density function (PDF) shown below: 000 00000 (0 ≤x≤3) (otherwise) Find the probability that the component will fail between 2 and 3 years. TOONOOTONTO E 0.3 0.2 R 0.1 T U 0.5 PDF IN 2.5 P(2 ≤x≤ 3) = (b) A system includes four components (A, B, C and D), one of which will fail over a time period. The probabilities of the mutually exclusive component failures are P(A) = 0.03 P(B) = 0.15 P(C) = 0.08 P(D)=0.12 The probability of a system failure (F) given each component failure is P(F|A) = 0.25 P(FB) = 0.20 P(F|C) = 0.30 P(FD)= 0.15 Find the total probability, P(F), that the system will fail in the time period.
j W f(x) = (x² /9 20 1px (a) The failure time in years (x) of a component has the probability density function (PDF) shown below: 000 00000 (0 ≤x≤3) (otherwise) Find the probability that the component will fail between 2 and 3 years. TOONOOTONTO E 0.3 0.2 R 0.1 T U 0.5 PDF IN 2.5 P(2 ≤x≤ 3) = (b) A system includes four components (A, B, C and D), one of which will fail over a time period. The probabilities of the mutually exclusive component failures are P(A) = 0.03 P(B) = 0.15 P(C) = 0.08 P(D)=0.12 The probability of a system failure (F) given each component failure is P(F|A) = 0.25 P(FB) = 0.20 P(F|C) = 0.30 P(FD)= 0.15 Find the total probability, P(F), that the system will fail in the time period.
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:(c) If a system with the four independent components is configured as shown below, find the
reliability of the system, P(R)=1-P(F) (i.e. the probability of non-failure). Note that the
conditional probabilities used in (b) do not apply here.
Failure probabilities:
P(A) = 0.03
P(B) = 0.15
IBI
1
BI
|▬▬▬▬▬▬
(d) Repeat (c) for the configuration below.
D.
|
C-------
C
|
P(C) = 0.08
P(R) =
P(R) =
P(D) = 0.12
Enter
hift
Transcribed Image Text:j
6.
W
f(x) = (x²/9
Lo
(a) The failure time in years (x) of a component has the probability density function (PDF)
shown below:
B
1
(0 ≤ x ≤3)
(otherwise)
Find the probability that the component will fail between 2 and 3 years.
0.9
0.8
0.7
0.6
0.5
0.4
0.3
E
0.2
0.1
R
O
T
0.5
PDF
P(2≤x≤3)=
(b) A system includes four components (A, B, C and D), one of which will fail over a time
period. The probabilities of the mutually exclusive component failures are
P(A) = 0.03 P(B) = 0.15
P(C) = 0.08
P(D) = 0.12
2.5
The probability of a system failure (F) given each component failure is
P(F|A) = 0.25
P(FB) = 0.20
P(F|C) = 0.30
P(FD)= 0.15
Find the total probability, P(F), that the system will fail in the time period.
Total Probability
P(E)=P(4)P(E|A₂)+ P(A₂) P(E | A₂)+...+ P(4)P(E\A)
P(F) =
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