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.