A sensor that detects dangerous levels of carbon monoxide (CO) in industrial app

Question

A sensor that detects dangerous levels of carbon monoxide (CO) in industrial applications is being tested. The manufacturer claims that it gives an alarm in 98% of such dangerous situations. However, it also responds to hydrogen, alarming 50% of the time when significant levels are present, and sometimes it gives an erroneous alarm in 0.4% of situations when neither carbon monoxide nor hydrogen are present. Dangerous levels of CO are present in 8% of situations in a particular industry. Significant levels of hydrogen are present in 5%. You may assume that CO and hydrogen never occur together (although in practice this is not true).

A) What is the probability that the sensor's alarm will trigger? Enter your answer to FOUR decimal places (0.XXXX). (Answer is NOT 0.0183))

B) Referring to the CO sensor question, what is the probability that dangerous levels of carbon monoxide are present if the alarm goes off? Enter your answer to FOUR decimal places (0.XXXX). Answer is NOT (0.0002))

C) Referring to the CO sensor question, what is the probability that there are dangerous levels of carbon monoxide if the alarm does not go off? Enter your answer to FOUR decimal places (0.XXXX).

Answer is NOT (0.6606)

Explanation / Answer

Let CO shows the event that carbon monooxide present, H shows the event that hydrogen present and N shows the event that neither carbon mono-oxide not hydrogen present. Let A shows the event that sensor gives alarm and S shows the event that sensor does not give alarm. So we have

P(CO) = 0.08, P(H) = 0.05, P(N) = 1- P(CO) -P(H) = 1-0.08 -0.05 = 0.87

And we have

P(A|CO) = 0.98, P(A|H) = 0.50, P(A|N) = 0.004

(A)

By the law of total probability the probability that the sensor's alarm will trigger is

P(A) = P(A|CO)P(CO) + P(A|H)P(H)+P(A|N)P(N) = 0.98*0.08 + 0.50*0.05 + 0.004*0.87 = 0.10688

Answer: 0.1069

(B)

Here we need to find the probability P(CO|S)=?

From the complement rule we have

P(S) = 1 - P(A) = 1- 0.10688 = 0.89312

And

P(S|CO) = 1- P(A|CO) = 0.02

So the requried probability is

P(CO|S) = [P(S|CO)P(CO)] / P(S) = [ 0.02 * 0.08] / 0.89312 = 0.0018

Answer: 0.0018

(C)

Here we need to find the probability P(CO|A)=?

The required probability is

P(CO|A) = [P(A|CO)P(CO)] / P(A) = [ 0.98 * 0.08] / 0.10688 = 0.7335

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