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Gnoma [55]
3 years ago
7

A new test to detect tb has been designed. it is estimated that 89% of people taking this test have the disease. the test detect

s 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?
Health
2 answers:
GaryK [48]3 years ago
5 0
The probability is 1 in 945 million
aleksley [76]3 years ago
5 0

Answer:

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 P(A/B) as the conditional probability '' The probability of the event A given that we know that the event B occurred'' as :

P(A/B)=\frac{P(A,B)}{P(B)}

Where P(B)>0

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

P(A)=1-P(A')

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(D)=0.89

P(D')=1-P(D)=1-0.89=0.11

P(D')=0.11

P : ''The test is positive''

P(P/D)=0.97

P(P'/D)=1-P(P/D)=1-0.97=0.03

P(P'/D)=0.03

And P(P'/D')=0.99

We are looking the probability of P(P')

P(P') = P [(P'∩D) ∪ (P'∩D')]

Given that this events are independent between them :

P(P')=P(P',D)+P(P',D') (I)

Let's write the conditionals for this problem :

P(P'/D)=\frac{P(P',D)}{P(D)} ⇒

P(P',D)=P(P'/D).P(D)

P(P',D)=(0.03).(0.89) (II)

And the another conditional :

P(P'/D')=\frac{P(P',D')}{P(D')} ⇒

P(P',D')=P(P'/D').P(D')

P(P',D')=(0.99).(0.11) (III)

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

P(P')=(0.03).(0.89)+(0.99).(0.11)

P(P')=0.1356

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

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