Question
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Difficulty Level:
4
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Suppose that 5% of people of your age and heredity have diabetes. Suppose that a
blood test has been developed that correctly gives a positive test result in 80%
of people with diabetes, and gives a false positive in 20% of the cases of people
without diabetes. Suppose you take the test, and it is positive. What is the probability
that you actually have diabetes, given the positive test result?
Solution
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Explanation Quality:4
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This questions is pretty forward. You can expect such questions if you are interviewing
for financial firms such as Goldman Sachs, SIG et al. Let this be a refresher on
basic probability too.
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Let's denote Probability that I have diabetes as P(D)
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=
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0.05
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Let's denote Probability that I don't have diabetes as P(-D)
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=
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1-0.05 = 0.95
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Probability my test is + given that I have diabetes
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=
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0.80
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Probability my test is + given that I don't have diabetes
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=
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0.2
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Note that this is a conditional probability question And here are some useful formulas
that we would use.
Baye's Theorem: P(A|B) = P(A and B) / P(B)
Total Probability Theorem: P(A) = P(A|B)*P(B) + P(A|-B)*P(-B)
We are actuall required to find P(D|+) i.e. I have diabetes given that my test is
positive. Caution ! P( D|+) IS NOT EQUAL TO P(+|D)
Using Baye's theorem I can write:
P(D|+) = P(+|D) * P(D) / P(+)
Setting up this equation pretty much solves the problem. Note we already know P(D)
= 0.05 and P(+|D) = 0.80.
All we need to know to crack this question is P(+) which we can using the total
probability theorem as follows:
P(+) = P(+|D)*P(D) + P(+|-D)*P(-D)
P(+) = 0.8*0.05 + 0.2*0.95
P(+) = 0.23
Finally we find P(D|+)
P(D|+) = P(+|D) * P(D) / P(+)
P(D|+) = 0.8 * 0.05 / 0.23
P(D|+) = 0.17
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