Despite our ability to manipulate our realities with belief, will, magic, prayer, psychic intuition the hard real real is below. It is a good base point to start to determine the logic of any situation. Simply being aware that such a process exists should give us pause before we conclude anything and we should also realize that historical evidence is very important, as well.
If you estimated the probability to be between 70%-80% you’re in good company. Study after study shows that when doctors are given this problem (with the language of breast cancer and mammogram replacing the language of aliens and shapeshifter-detecting apps) about 6 in 7 of them give an estimate in the 70%-80% range. The correct answer is 7.76%. In other words, even though the accuracy of the test is 80% a positive test result means only 7.76% chance of the individual being an alien (or of having breast cancer, if the problem si couched in that language). If you want to see how the numbers stack up, the below applies the Bayes formula with the numbers given in the scenario.
Bayes and the Shapeshifter Problem:
Bayes’ theorem:
This is one of the most important equations in science.
P(H|E = P(E|H)P(H) / P(E)
- P(H|E) is called posterior probability of a hypothesis given some evidence.
- P(H) is prior probability based on evidence
- P(E|H) is the likelihood of the event without consideration of circumstances
- P(E) is the probability or (value) of the evidence
We want to know P(H|E)—the probability of the the alien shapeshifter hypothesis being true given the evidence of a positive test result.
What we are told is the likelihood P(E|H)—the probability of a positive test occurring given the individual is an alien, which in this case is 80%. Bayes tells us that we also need to take into account P(H), which is the chance that an individual chosen at random is an alien—1% in this case. Furthermore we also need to take into account P(E), which is the chance of getting any positive test—either a real positive or a false positive, in this case the chance of a real positive is 1% x 80% (aliens times accuracy) or 0.008. The chance of a false positive is 99% x 9.6%. (not aliens times fake aliens ‘humans’) most of them can actually be found on the internet (me) or, in other words, 0.09504). So P(E) = 0.008 + 0.09504 or 0.10304.
After applying the Bayes formula we see the result of P(H|E) is 7.76%