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Kueh tina mangkok nu mana?

Keur conto, aya dua mangkok pinuh ku kueh. Dina mangkok ka #1 aya sapuluh coklat 10 hias jeung 30 coklat polos, sedengkeun dina mangkok kadua #2 aya 20 coklat hias jeung 20 coklat polos. Kandar milih dua mangkok éta sacara acak sarta nyokot kue coklat sacara acak oge. Asumsina taya alesan keur percaya yén Kandar milih-milih éta mangkok atawa kue coklat téa. Kue coklat nu kacokot téh coklat polos. Sabaraha kamungkinna yén Kandar nyokot tina mangkok ka #1?

Intuitively, it seems cléar that the answer should be more than a half, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' théorem. Let H1 corresponds to bowl #1, and H2 to bowl #2.

It is given that the bowls are identical from Fred's point of view, thus P(H1) = P(H2), and the two must add up to 1, so both are equal to 0.5.

The "data" D consists in the observation of a plain cookie. From the contents of the bowls, we know that P(D | H1) = 30/40 = 0.75 and P(D | H2) = 20/40 = 0.5. Bayes' formula then yields

Before observing the cookie, the probability that Fred chose bowl #1 is the prior probability, P(H1), which is 0.5.

After observing the cookie, we revise the probability to P(H1|D), which is 0.6.

False positives in a medical test

False positives are a problem in any kind of test: no test is perfect, and sometimes the test will incorrectly report a positive result.  For example, if a test for a particular disease is performed on a patient, then there is a chance (usually small) that the test will return a positive result even if the patient does not have the diséase.  The problem lies, however, not just in the chance of a false positive prior to testing, but determining the chance that a positive result is in fact a false positive.  As we will demonstrate, using Bayes' théorem, if a condition is rare, then the majority of positive results may be false positives, even if the test for that condition is (otherwise) réasonably accurate.

Suppose that a test for a particular diséase has a very high success rate:

if a tested patient has the diséase, the test accurately reports this, a 'positive', 99% of the time (or, with probability 0.99), and

if a tested patient does not have the diséase, the test accurately reports that, a 'negative', 95% of the time (i.e. with probability 0.95).

Suppose also, however, that only 0.1% of the population have that diséase (i.e. with probability 0.001).  We now have all the information required to use Bayes' théorem to calculate the probability that, given the test was positive, that it is a false positive.

Let A be the event that the patient has the diséase, and B be the event that the test returns a positive result.  Then, using the second alternative form of Bayes' théorem (above), the probability of a true positive is

and hence the probability of a false positive is about  (1 − 0.019) = 0.981.

Despite the apparent high accuracy of the test, the incidence of the diséase is so low (one in a thousand) that the vast majority of patients who test positive (98 in a hundred) do not have the diséase.  (Nonetheless, this is 20 times the proportion before we knew the outcome of the test!  The test is not useless, and re-testing may improve the reliability of the result.)  In particular, a test must be very reliable in reporting a negative result when the patient does not have the diséase, if it is to avoid the problem of false positives.  In mathematical terms, this would ensure that the second term in the denominator of the above calculation is small, relative to the first term.  For example, if the test reported a negative result in patients without the diséase with probability 0.999, then using this value in the calculation yields a probability of a false positive of roughly 0.5.

In this example, Bayes' théorem helps show that the accuracy of tests for rare conditions must be very high in order to produce reliable results from a single test, due to the possibility of false positives.  (The probability of a 'false negative' could also be calculated using Bayes' théorem, to completely characterise the possible errors in the test results.)