^{®}pregnancy test. They advertise it as 99% accurate. Now, this actually means that 99% of the time, if there is a sufficient concentration of a particular hormone, the mechanism will indicate a pregnancy. What I can't locate, however is the rate of false positives- or, what percentage of the time an insufficient concentration of said hormone will indicate a pregnancy. This is critical information, because the actual probability of

__getting__a false positive is not very intuitive. The way you actually calculate the probability that you're pregnant given a positive result on a pregnancy test is to use conditional probabilities, or Bayes' Theorem. We say that P(A|B) is the probability of A given B. So, the probability of being pregnant (call it preg) given that you observe a positive result (call it +) on a pregnancy test can be written P(preg|+). We calculate this probability... but to do so, we need the "accuracy," as Pfizer reports it. They observe a positive result (+) given a pregnancy (preg) with 99% probability. So P(+|preg) = 0.99. We also need to know what the chance that you're pregnant is. In the U.S., there are about 14 births/1000 population (according to the CIA Factbook). Given that we have about 300M people, approximately 50% of whom are women, we can say that if you are a woman, your chance of being pregnant is approximately 2%. This is, of course, a rough estimate, but bear with me. That is, P(preg)=0.02. Lastly, we need one piece of information I can't seem to find: what is the probability of getting a positive result given no pregnancy, or P(+|not preg)? Well, if we assume there's a 1% chance of such a reading, P(+|not preg)=0.01. Now, the formula is: P(preg|+)=P(+|preg)P(preg)/P(+), where P(+) = P(+|preg)P(preg)+P(+|not preg)P(not preg). We can now calculate! So, P(preg|+) = 0.99*0.02/(0.99*0.02+0.01*0.98)=0.67. Under these assumptions, there's only a 67% chance that you really are pregnant given you got a positive reading on the test! Holy crap! So from what I can tell, then, the false positive rate had better be reeeeallly low if you want to trust your tests.

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