7. A medical test for a disease has a chance of 70% of showing positive for an i

Question

7. A medical test for a disease has a chance of 70% of showing positive for an individual if he/she actually
has the disease. For an individual not having the disease it has a 10% chance of showing positive. Suppose
only 1% of the population actually has the disease. You pick an individual uniformly at random from
the population. The chance that the test shows positive for this individual is
A. 0.8034 B. 0.4000 C. 0.1060 D. 0.2104 E. 0.700

I know the answer is C, but have no idea how to work it out.

Explanation / Answer

This is a Bayes' Rule Problem; the best way to solve it is to break the population up into specific outcomes and find the probability of each outcome.

.99=Prob. that given individual doesn't have disease
>Within that 99%, in 10% the test is positive, so that's .99*.1=.099=Prob. of non-diseased person testing positive
>Within that 99%, 90% (100%-10%) test negative, so .99*.9=.891=Prob. of non-diseased person's test being negative

.01=Prob. that given individual has disease
>Within that 1%, 70% test positive, so that is .01*.7=.007=Prob. of diseased person testing positive
>Within that 1%, 30% (100%-70%) test negative, so .01*.3=.003=Prob. of diseased person testing negative

As this encompasses all possible outcomes, we can simply add the probabilities for both situations in which a person tests positive, which is .099+.007=.106=Answer C

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