A terrible disease sweeps around the world, luckily only affecting 1

Question

CHALLENGE QUESTIONS

A terrible disease sweeps around the world, luckily only affecting 1 in 10,000, but for that one, the disease is lethal. Shortly after the disease was discovered, scientists developed a test that is 99% accurate regardless of whether you have the disease. In other words, the test yields the correct positive or correct negative result 99% of the time. You take the test and a week later, you receive the lab report. The outcome of the test is positive. What is the probability you have the disease?

A.  \frac{99}{1,000,000}

B.  \frac{1}{102}

C.  \frac{1}{100}

D.  \frac{10,099}{1,000,000}

E.  \frac{99}{100}

chetan2u · Accepted Answer

Hi...

It is a tough Q if you don't understand the wordings and is ofcourse TRICKY and that is why none have got the correct answer above.

The result is POSITIVE..
Two cases..
1) Correct:- 99%
So he is 1 out of 10000.
Probability=  \frac{1}{10000}*\frac{99}{100}=\frac{99}{1000000} ..
2) Wrong:- 1%
So he is out of 9999..
Probability=  \frac{9999}{10000}*\frac{1}{100}=\frac{9999}{1000000} ..

So PROBABILITY that it is correct = (99/1000000)/(99/1000000+9999/1000000) = \frac{99}{99(1+101)}=1/102 
B

IanStewart · Answer

Take a million people, and say they all get tested.

1,000,000 people

100 have the disease
999,900 do not

of these:
99 test positive for the disease *and* have the disease
1 tests negative for the disease *and* has the disease
9999 test positive for the disease *and* do not have the disease
989901 test negative for the disease *and* do not have the disease

So for every million people, 10,098 will test positive. Only 99 will have the disease. If you test positive, you're one of these 10,098 people, and there is a 99/10,098 = 1/102 chance you actually have it.

mdeakubahmed · Answer

Should be .99

I do not see any catch in this question.. I might be missing something.

IanStewart · Answer

If you want to see the catch, think of more extreme numbers:

Say 1 person in the world has disease ZZZ, and 10 billion people do not. There's a test that is 50% reliable and you take it. If you test "positive", is there a 50% chance that you have the disease? Surely not; billions of people will test positive who don't have ZZZ.

There is a profound difference between the following two questions:

If you have the disease, what is the probability you test positive? (the answer is 99%)

If you test positive, what is the probability you have the disease? (the answer is not 99%)

The original post asks the second question, not the first.

hellosanthosh2k2 · Answer

Great question on conditional probability.

P(A happening / given B occured) = P(A AND B)/P(B)

P(You have disease/given outcome is positive) = P(you have disease AND outcome is positive)/P(outcome is positive)

P(outcome is positive) = false positive + true positive =  (9999/10000)  *  (1/100)  +  (1/10000)  *  (99/100)

P(you have disease AND outcome is positive) =  (1/10000) * (99/100)

So P(A given B) = P(You have disease/given outcome is positive) =  (1/10000) * (99/100)  / ( (9999/10000)  *  (1/100)  +  (1/10000)  *  (99/100) )

=  99 / (9999 + 99)  
 
 = 1/ (1 +  9999/99 ) = 1 / (1 + 101) = 1/ 102

Answer (B)

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A terrible disease sweeps around the world, luckily only affecting 1

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