Question

Utilising the following assumed reliabilities:

Smoke detection system 95%

Automatic sprinkler system 95%

Smoke exhaust system 85%

Fire alarm system 98%

1. Calculate the system probability that at least three of the four fire systems operate, provide enough significant digits especially when the answer is very close to zero or one.

Answer 1

The probability that at least 3 of the 4 systems operate is

= The probability that 3 systems operate + The probability that all the 4 systems operate

Three of the 4 systems can operate in 4 ways:-

i) Smoke detection, automatic sprinkler, and smoke exhaust systems operate, the fire alarm system doesn't operate.

P = 0.95 * 0.95 * 0.85 * (1-0.98) = 0.0153

ii) Smoke detection, automatic sprinkler, and fire alarm systems operate, the smoke exhaust system doesn't operate.

P = 0.95 * 0.95 * (1-0.85) * 0.98 = 0.1327

iii) Smoke detection, smoke exhaust and the fire alarm systems operate, the automatic sprinkler system doesn't operate.

P = 0.95 * (1-0.95) * 0.85 * 0.98 = 0.0396

iv)  Automatic sprinkler, smoke exhaust and the fire alarm systems operate, the Smoke detection system doesn't operate.

P = (1-0.95) * 0.95 * 0.85 * 0.98 = 0.0396

So, the probability that 3 of the 4 systems operate is

= 0.0153 + 0.1327 + 0.0396 + 0.0396

= 0.2272

The probability that all of the 4 systems operate is

= 0.95 * 0.95 * 0.85 * 0.98 = 0.7518

Hence, the total probability that at least 3 of the systems work is

= 0.2272 + 0.7518

= 0.9790

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