If you test one person at random, the odds of that person being the President equals the probability of a true positive divided by the probability that the test

If you test one person at random, the odds of that person being the President equals the probability of a true positive divided by the probability that the test provides a Positive result (i.e. the sum of the probabilities for True and False Positives). The probability of a true positive is the probability that test subject is the President times the conditional probability that the subject tests positive given that they are the President. If the subject is chosen at random from the population of the US, then the a priori odds of this person being the President is about 1/314M times a conditional probability of testing positive of 1. The probability of a false negative is the a priori odds of a random person being the President, times the conditional probability of a non-President testing positive. By hypothesis, this is (1-1/314M)0.00001. So the odds of a subject chosen at random from the overall US population must be approximately (1/314M)/((1/314M)+(1-1/314M)0.00001), which simplifies to 1/(1+(314M-1)*0.00001, or about 1/3141. The ~99.97% chance that the test result is a false positive for a subject chosen at random is a consequence of the prior probability of that subject being the President in the first place being over 3000 times lower than the probability of a positive test result. In other words, though the sensitivity of the test is perfect, the specificity of the test is insufficient to isolate a condition as rare as the Presidency. It has nothing to do with testing one person being inherently more decisive than testing more than one. Only narrowing the pool of candidates in a way which will increase the prior probability that the subject is the President will improve the situation, not random decimation to a single test subject. The GP gave several correct examples such as limiting the test to those who were much more likely to be the President in the first place (eg. a person randomly found sitting in the President's chair at the Oval Office might have a prior probability of being the President of greater than 0.01, millions of times greater than that of a randomly selected person in the US).