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A terrible disease sweeps around the world, luckily only [#permalink]

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05 Sep 2008, 16:39

This topic is locked. If you want to discuss this question please re-post it in the respective forum.

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 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?

Last edited by freestyla86 on 06 Sep 2008, 07:12, edited 1 time in total.

Re: Probability Word Problem (Not Multiple Choice) [#permalink]

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05 Sep 2008, 17:56

freestyla86 wrote:

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 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?

Let me try

Probability that one has this disease means that one has to have the disease and the test he/she has taken must be correct positive.

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 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?

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

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.
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Re: Probability Word Problem (Not Multiple Choice) [#permalink]

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06 Sep 2008, 00:21

IanStewart wrote:

freestyla86 wrote:

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 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?

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.

Brilliant!
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----------------------------------------------------------- 'It's not the ride, it's the rider'

Re: Probability Word Problem (Not Multiple Choice) [#permalink]

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06 Sep 2008, 05:22

I beg to disagree with this..

These two are independent cases,

1 ) affecting 1 in 10,000; so P1= 1/10,000 => so if somebody asks, what is the probability of anybody getting effected , asswer is 1/10000

2 ) test that is 99% accurate => so if somebody is tested, and whatever the result be, the accuracy of the result is =P2=99/100

Here, you take the test and the results are out , so do not need to worry about the 1st case at all. The question is not asking : what is the probability that if a person is tested, the tested person will be +ve, which will be 1-P1*P2=1-9999/10000*99/100

Re: Probability Word Problem (Not Multiple Choice) [#permalink]

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06 Sep 2008, 07:10

My take is :

The outcome of the experiment independent of whether the person has the disease or not . SO the probalility of the the experiment being correct is independent of the probabality of a person being effected by the desase.

Re: Probability Word Problem (Not Multiple Choice, No OA) [#permalink]

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06 Sep 2008, 07:54

freestyla86 wrote:

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 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?

I will try to solve this one in a long way.

Assuming 1 million people in the city.

People who have disease = (1/10,000) x (1,000,000) = 100 persons People who do not have disease = 1,000,000 - 100 = 999,900 persons

For people with disease, the lab can identify correctly = (99/100)*100 = 99 persons the lab will make a mistake on = (1/100)*100 = 1 person

For people with no disease, the lab can identify correctly = (99/100) * 999,900 = 989,901 person the labe will make a mistake on = (1/100)*999,900 = 9,999 person

Now, Prob(you have disease based on the fact that the lab say so) = (99)/(99 + 9,999) = 99 / 99*(1+101) = 1/102 = 0.9804%

Last edited by devilmirror on 06 Sep 2008, 13:00, edited 4 times in total.

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.
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Re: Probability Word Problem (Not Multiple Choice, No OA) [#permalink]

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06 Sep 2008, 13:01

Yes this is a conditional probability problem which needs to be solved using Bayes' theorem. Getting a little more technical, we know the priory P(D) and and marginal probability P(p) (all possibilities of getting positive test result). We need to find the posterior P(D|p).

D=having disease and ~D=not having disease p=testing positive and n= testing negative

The question asks probability P(having disease given positive result)