For many people, believer and skeptic alike, the answer to the question "Is the Resurrection of Jesus improbable?" is patently obvious: "Of course it's improbable!" And in a sense they're right. By definition, miracles such as the resurrection of Jesus do not occur often, if they occur (or have occurred) at all, so that by a frequency interpretation of probability, at least, the resurrection is unquestionably improbable. Indeed, with the use of this restricted frequentist understanding of probability some powerful arguments can be raised against the resurrection.
Doubtless the most famous such argument is David Hume's, including his "general maxim": "...that no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish.” Per Hume's reasoning, miracle claims run up against a wall of improbability so high that it only be overcome by evidence so great that it somehow demonstrates the probability of the miracle to be even higher. But testimonial evidence can never clear the wall, because human testimony is inherently, demonstrably corruptible. Thus, so the argument goes, for any given miracle claim it's always more probable that the claimant is simply lying (or deluded, or hallucinating, etc.) than it is that the laws of nature have failed to hold. Extraordinary claims require extraordinary evidence.
Hume's argument appears eminently rational. Many observers consider it an informal expression of Bayes' Theorem, a powerful mathematical tool for assessing and updating probabilities in light of available evidence and prior probabilities. In principle we can use Bayes' Theorem to assess the probability of the resurrection, where R is the hypothesis that Jesus rose from the dead, E is the proposed evidence, and K is our background knowledge, which includes all relevant but not case-specific information:
P(E│R & K) x P(R│K)
P(R│E & K) = --------------------------
P(R│E) here means basically "the probability that the resurrection occurred, given the evidence provided in support of it." Per Bayes' Theorem, then, the probability of R given evidence E equals the likelihood of R on E (or the explanatory power of R) times the prior probability of R (a somewhat subjective probability estimated prior to, or apart from, the specific evidence in question). Thus for the resurrection of Jesus to be "probable" the explanatory power of the resurrection conjoined with its prior probability must exceed the probability of simply having the evidence for it that we do have apart from any particular hypothesis.
Now I am not a mathematician or statistician. So to appreciate the usefulness of the theorem I find it helpful to examine the likelihood ratio and the prior probability of R separately, with the proviso that their probabilities will be diminished when multiplied together. First let's consider likelihood, or the explanatory power of R, leaving out P(R│K) for the moment:
P(E│R & K)
For R to enjoy substantial explanatory power, the probability of our having the evidence E on the resurrection hypothesis must be high relative to the probability that we would have E regardless. In the spirit of Hume, many philosophers have argued that even if P(E│R & K) is extremely high, it cannot be all that much higher than P(E│K), simply because given human nature we could expect to have something much like E (stories about men rising from the dead) whether the resurrection occurred or not.
I don't think that's the best way to read the evidence. Yes, people (not just religious devotees but scientists and skeptics) have been known to make up stories about all sorts of things. But the set of facts bearing on the case of the resurrection narrative takes place in a specific historical context, involving identifiable historical personalities (e.g., Jesus, Pilate, Herod, Joseph of Arimathea, Mary Magdalene, and Peter), geographic references and topographical features, and contemporaneous events. Those events include the birth of the church in Jerusalem, on the preaching of the resurrection, despite violent persecution, and a scant few weeks after the crucifixion; and the conversion of Saul of Tarsus. Additionally the "criterion of embarrassment" suggests low intrinsic probability of evidence such as the reports of women being the last to leave Jesus' side and the first to discover his tomb empty. In other words the facts bearing on the case extend far beyond the mere "testimony" of the Gospel writers and the apostle Paul. The probability of all that evidence taken together, had Jesus not risen from the dead, must be extremely low; that being the case the explanatory power of the resurrection hypothesis is proportionately high.
Next consider P(R│K), the prior probability of the resurrection. To hear skeptics, this number must be "vanishingly small," in that the resurrection runs counter to the laws of nature. But a couple of observations weaken that suggestion. First, the custodians of the scientific enterprise, those who inform us about the laws of nature, are human beings. By Hume's own estimation, merely being people is enough to make scientists and skeptics susceptible to dishonesty and delusion. Second, given the history of science, in many cases we cannot be entirely certain just what the laws of nature actually are or what they forbid (let alone how well they hold up against supernatural power). Quantum mechanics and general relativity, for example, are two of the best-confirmed scientific theories of our time, yet at certain points they become incompatible; so the search for a coherent theory of quantum gravity continues.
Men do not often rise from the dead. But neither does life often originate on earth. Life emerges only as it reproduces "after its kind," as Genesis memorably says it. On a frequency interpretation of probability the laws that prevent life from originating on earth would have to be just as inviolable as the laws that prevent men from rising from the dead (neither event has been confirmed in modern times). Yet here we are. Perhaps miracles are not so "improbable" after all. Indeed if God exists and decides to perform a miracle, the prospect of a miracle becomes neither probable nor improbable, but absolutely certain. It seems reasonable to suggest, then, that we are in no position to determine just how probable miracles or resurrections are generally. In that case it seems to me (though I'm not committed to this) that we could leave out prior probability and merely assess the explanatory power of the resurrection, which as I have argued is quite high.
Despite all this the skeptical approach is typically to tear the resurrection out of its historical context and restate it as simply "the probability of a man rising from the dead." Granted, the probability of you, or me, or the average man on the street rising from the dead would have to be negligible. But the facts of the case have to do with a man who claimed exclusive divine authority, who convinced multitudes that he was the Son of God, who was widely reported performing miracles of healing and exorcism, who predicted his own crucifixion and resurrection, and finally, who somehow disappeared from his tomb. However improbable a resurrection may be generally, clearly the resurrection of Jesus of Nazareth is much more probable than that of most men. If that is true, then Jesus' resurrection cannot be, as some philosophers imagine, "as improbable as anything possibly can be." At worst it leaves the issue unresolved, in which case Paul's question to King Agrippa at Caesarea remains pertinent today:
"Why should it be thought incredible… that God raises the dead?" (Acts 26:8)