Anew test to detect tb has been designed. it is estimated that 89% of people taking this test have the disease. the test detects the disease in 97% of those who have the disease. the test does not detect the disease in 99% of those who do not have the disease. if a person taking the test is chosen at random, what is the probability of the test indicating that the person does not have the disease?

0.1356

Explanation:

This is a probability exercise. Let's define some probability concepts.

Given two events A and B :

(A∩B) = (A,B)

(A,B) is the intersection event where A and B occur both at the same time.

We define as the conditional probability '' The probability of the event A given that we know that the event B occurred'' as :

Where

Now, if A is an event and is its complement ⇒

Finally we define the probability of the union between two events A and B :

P(A∪B) = P(A) + P(B) - P(A,B)

If the events A and B are independent between them ⇒ P(A,B) = 0 ⇒

P(A∪B) = P(A) + P(B)

Let's define the following events for this exercise :

D : ''People taking this test that have the disease''

P : ''The test is positive''

And

We are looking the probability of

P(P') = P [(P'∩D) ∪ (P'∩D')]

Given that this events are independent between them :

(I)

Let's write the conditionals for this problem :

⇒

(II)

And the another conditional :

⇒

(III)

Replacing (II) and (III) in (I) :

We find that the probability of the test indicating that the person does not have the disease ( P(P') ) is 0.1356

                                                        water

a colorless, transparent, odorless, tasteless liquid that forms the seas, lakes, rivers, and rain and is the basis of the fluids of living organisms.

                                                        halon  

any of a number of unreactive gaseous compounds of carbon with bromine and other halogens, used in fire extinguishers, but now known to damage the ozone layer.

                                                  dry chemical

extinguishing agent used on class a, class b, and class c fires.

                                                  carbon dioxide

a colorless, odorless gas produced by burning carbon and organic compounds and by respiration. it is naturally present in air (about 0.03 percent) and is absorbed by plants in photosynthesis.

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