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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

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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? Select the correct answer from above options

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Correct Answer - (i) `1/400` (ii) `19/200` Let `E_(1)=` event of availability of the first engine. And, `E_(2) =` event of availability of the second engine. Then, `P(E_(1))=P(E_(2))=0.95` and `P(bar(E)_(1))=P(bar(E)_(2))=(1-0.95)=0.05`. (i) P( neither of them is available when needed) `=P (bar(E)_(1) and bar(E)_(2))=P(bar(E)_(1))xxP(bar(E)_(2))`. (ii) P( an engine is available when needed) `=P [(E_(1)" and not "E_(2)) or (E_(2)" and not "E_(1))]` `=P[(E_(1) and bar(E)_(2)) or (E_(2) and bar(E)_(1))]` `=P (E_(1) nn bar(E)_(2))+P(E_(2) nn bar(E)_(1))` `={P (E_(1))xxP(bar(E)_(2))}+{P(E_(2))xxP(bar(E)_(1))}`.

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