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A town has two fire-extinguishing engines, functioning independently. The probability of availability of each engine when needed is 0.95. What is the probability that 

(i) neither of them is available when needed? 

(ii) an engine is available when needed?

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Given: Let A and B be two fire extinguishing engines . 

The probability of availability of each of the two fire extinguishing engines is given i.e., 

P(A) = 0.95 and P(B) = 0.95 

\(\Rightarrow\) P(\(\overline A\)) =  0.05 and P(\(\overline B\)) = 0.05

To Find: 

(i) The probability that neither of them is available when needed

Here, P(not A and not B) = P(\(\overline A\cap \overline B\))

= P(\(\overline A\)) x P(\(\overline B\))

= 0.05 x 0.05

= 0.0025 = \(\frac{1}{100}\)

Therefore, The probability that neither of them is available when needed is\(\frac{1}{400}\)

(ii) an engine is available when needed

Here, P(A and not B or B and not A) = P( A ∩ \(\overline B\)) + P(B ∩ \(\overline A\))

\(\frac{19}{200}\)

Therefore, The probability that an engine is available when needed is\(\frac{19}{200}\)

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