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# Willywutang and STDs

## Question

While willywutang was trying to be clever by using only two condoms to satisfy three women, increased friction between overlapped latex layers produces a tear and a dangerous accident! He then becomes very worried about having contracted an STD, especially since one of the women was 'looked like she could have an STD. Although a random person has only probability 0.001 of having an STD, poor willywutang just can't sleep over those odds. Frantically he hustles to the nearest drugstore, to purchase the ACME All-Purpose STD Checker. The packaging boasts a 0.93 correctness probability. That is, if the user has an STD, the ACME STD Checker will return positive 93% of the time; if the user does not have an STD, it will return negative 93% of the time. Willywutang returns home and uses the checker in his bathroom.
To his dismay, the results are positive.
Assuming that willywutang's promiscuity on average is identical to that of a randomly chosen person, what is the probability that willywutang has STDs?

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The positive result of the test is possible by 2 ways:
Someone "X" has the STD and the checker gave the right result or X does not have STD and the checker gave the wrong result.

Random probability that X has STD = 0.001
Prob that the test gave correct result = 0.93

So, in a sample population, number of correct positive results would be = 0.93 * 0.001 = 0.0093

Random probability that X does not have STD = 0.999
Prob that the test gave wrong result = 0.07 (showed positive for an actually negative person)

In the sample population, number of false positive results would be = 0.999 * 0.07 = 0.06993

So, for every one correct positive there are = 0.06993/0.0093 = 75.19 false positive results

So, the probability that W has STD given that the result shows positive is = 1/(1+75.19) = 0.013

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