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A carbon monoxide detector system uses two sensors, A and B. If carbon monoxide ispresent, there is a 98% chance that sensor A will detect it, a 94% chance that sensor Bwill detect it, and a 93% chance that both sensors will detect it. If carbon monoxide ispresent, what is the probability that it will be detected?

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EXPLANATION

Let name event A to the probability that the sensor A detect the monoxide and event B to the probability that the sensor B detects the monoxide.

We need to compute the probability that both sensors will detect it at same time:

[tex]P(A+B)=\lbrack P(A)-P(A\cap B)\rbrack+\lbrack P(B)-P(A\cap B)\rbrack+P(A\cap B)[/tex]

Replacing terms:

[tex]P(A+B)=\lbrack0.98-0.93\rbrack+\lbrack0.94-0.93\rbrack+0.93[/tex][tex]P(A+B)=0.05+0.01+0.93=0.99[/tex]

The probability that the present monoxide would be detected is 0.99 or 99%

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