Question

1) A sprinkler system inside an office building has two types
of activation devices, D1 and D2, which operate independently. When
there is a fire, if either device operates correctly, the sprinkler
system is turned on. In case of fire, the probability that D1
operates correctly is 0.95, and the probability that D2 operates
correctly is 0.92. Assume the two activations devices operate
independently of each other.

a) Find the probability that both D1 and D2 will operate
correctly.

b) Find the probability that the sprinkler system will come
on.

c) Find the probability that the sprinkler system will
fail.

Answer #1

Given both D1 and D2 operates indepently.

P(D1) is the probability that D1 operates correctly = 0.95

P(D1^{c}) is is the probability that D1 does not
operates correctly = 1- P(D1) = 1-0.95 = 0.05

P(D2) is the probability that D2 operates correctly = 0.92

P(D2^{c}) is is the probability that D2 does not
operates correctly = 1- P(D2) = 1-0.92 = 0.08

a) P(both D1 and D2 operate correctly) = P(D1 D2) = P(D1)*P(D2) ( BESCAUSE THESE ARE INDEPENDENT EVENTS)

P(D1 D2) = 0.95 * 0.92 = 0.874.

c) The probability that the sprinkler system will fail = P(both
D1and D2 fail) = P(D1^{c}
D2^{c} )

P(D1^{c}
D2^{c} ) = P(D1^{c}) * P(D2^{c}) = 0.05 *
0.08 = 0.004

b) The probability that the sprinkler system will come on

P(sprinkler system will come on ) = 1 - probability of sprinkler
system will fail = 1 - P(D1^{c}
D2^{c} ) P(sprinkler system will come on ) = 1-0.004 =
0.996.

1- A particular automatic sprinkler system has two different
types of activation devices for each sprinkler head. One type has a
reliability of 0.87; that is, the probability that it will activate
the sprinkler when it should is 0.87. The other type, which
operates independently of the first type, has a reliability of
0.77. If either device is triggered, the sprinkler will activate.
Suppose a fire starts near a sprinkler head.
a) What is the probability that the sprinkler head...

An office building has two fire detectors. The probability is
0.015 that any fire detector of this type will fail to go off
during a fire. Find the probability that both fire detectors will
fail to go off in case of a fire. Assume that these two fire
detectors are independent of each other. Round your answer to four
decimal places.

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