The theorem itself can be stated simply. Beginning with a provisional hypothesis about the world (there are, of course, no other kinds), we assign to it an initial probability called the prior probability or simply the prior. After actively collecting or happening upon some potentially relevant evidence, we use Bayes’s theorem to recalculate the probability of the hypothesis in light of the new evidence. This revised probability is called the posterior probability or simply the posterior. Specifically Bayes’s theorem states (trumpets sound here) that the posterior probability of a hypothesis is equal to the product of (a) the prior probability of the hypothesis and (b) the conditional probability of the evidence given the hypothesis, divided by (c) the probability of the new evidence.

Consider a concrete example. Assume that you’re presented with three coins, two of them fair and the other a counterfeit that always lands heads. If you randomly pick one of the three coins, the probability that it’s the counterfeit is 1 in 3. This is the prior probability of the hypothesis that the coin is counterfeit. Now after picking the coin, you flip it three times and observe that it lands heads each time. Seeing this new evidence that your chosen coin has landed heads three times in a row, you want to know the revised posterior probability that it is the counterfeit. The answer to this question, found using Bayes’s theorem (calculation mercifully omitted), is 4 in 5. You thus revise your probability estimate of the coin’s being counterfeit upward from 1 in 3 to 4 in 5.

A serious problem arises, however, when you apply Bayes’s theorem to real life: it’s often unclear what initial probability to assign to a hypothesis. Our intuitions are embedded in countless narratives and arguments, and so new evidence can be filtered and factored into the Bayes probability revision machine in many idiosyncratic and incommensurable ways. The question is how to assign prior probabilities and evaluate evidence in situations much more complicated than the tossing of coins, situations like global warming or autism. In the latter case, for example, some might have assigned a high prior probability to the hypothesis that the thimerosal in vaccines causes autism. But then came new evidence — studies showing that permanent removal of the compound from these vaccines did not lead to a decline in autism. The conditional probability of this evidence given the thimerosal hypothesis is tiny at best and thus a convincing reason to drastically lower the posterior probability of the hypothesis. Of course, people wedded to their priors can always try to rescue them from the evidence by introducing all sorts of dodges. Witness die-hard birthers and truthers, for example.

McGrayne devotes much of her book to Bayes’s theorem’s many remarkable contributions to history: she discusses how it was used to search for nuclear weapons, devise actuarial tables, demonstrate that a document seemingly incriminating Colonel Dreyfus was most likely a forgery, improve low-resolution computer images, judge the authorship of the disputed Federalist papers and determine the false positive rate of mammograms. She also tells the story of Alan Turing and others whose pivotal crypto-analytic work unscrambling German codes may have helped shorten World War II.