Statistics can be very misleading if carelessly applied, for the magnitude of a probability depends greatly on our choice of assumptions. Thus, an experienced statistician can produce correct calculations, yet still deceive his audience who does not understand exactly what has been computed. For example, it is often said that a DNA match against a suspect proves guilt by odds of many millions to one, when in fact, such odds are meaningless since the probability of human error or contamination is many times greater than one in a million. Similarly, people are impressed by seemingly fantastic coincidences because they fail to consider all the different ways such a coincidence could occur. The latter is the case with the flawed statistical analysis of the Talpiot tomb inscriptions.
The real question we are trying to address probabilistically is: what are the odds that names matching that of the family of Jesus of Nazareth should appear in a random tomb, simply on the basis of the relative incidence of those names? The critical inscriptions are those of Yeshua bar Yehosef, Mariamne '(e) Mara, Yose, and Marya. These are interpreted as Jesus son of Joseph, Mary (Magdalene?), Joseph, and Mary. For the sake of argument, we will assume that there are two Marys in Jesus' family, although there is no solid evidence for such a claim, apart from the putative evidence we are considering here.
First, we will consider the probability of the incidence of each name. Yeshua/Yehoshua appears 71 out of 1986 times in Jewish funerary inscriptions, for a probability of 0.03575. Yose has a probability of 1/20, or 0.05, while Marya and its variants occur with a frequency of 1/4, or 0.25.
The flawed analysis uses a probability of 1/160 for the name Mariamne. While this is the correct frequency for that particular variation of Maryam/Marya, it ignores the fact that the investigators would have certainly regarded any variation of Maryam/Marya as a match for Mary Magdalene, so the appropriate probability to use is 1/4, as there is nothing in the name Mariamne, which is merely a variation of Maryam, to specify "Magdalene". The alternative name Mara is a variant of Martha, preceded by an aspirate ('), as a common shorthand for he kai, to precede a second name for people who went by two names.
So the probability of having four randomly selected tombs contain the names Yeshua, Yose, and two instances of Maryam/Marya equals: 0.03575 x 0.05 x 0.25 x 0.25 = 0.0001117, or about 1/10,000. However, this does not take into account the fact that the investigators had not four, but six ossuaries from which to choose possible matches. (There were ten in all, but only six had inscriptions.) Applying simple combinatorics, there are 6!/4!(6-4)! = 15 possible ways to select four out of six ossuaries (independent of the order in which they are chosen). Thus the probability ought to be multiplied by 15, yielding 0.0016755, or about 1/600 chance. Already, we have a probability sufficient to explain this as a random coincidence, since there are probably 1000 first-century tombs in Jerusalem alone (hundreds of which have been excavated), not counting the rest of ancient Israel.
Yet the probability is actually even higher than what we have stated. Since all names are gender-specific, each of the name frequencies contain a latent factor of 1/2 accounting for the probability of a person being male or female. For example, the name Maryam/Marya was held by 1/4 of all persons, but 1/2 of all women. Thus, each of the four name frequencies includes a "gender factor" of 1/2, which multiplies out to 1/16. This would imply that the probability of having two males and two females in our selected four ossuaries is 1/16, when such is obviously not the case. The actual probability is 6/16, since, as a matter of biological fact, men and women appear in close to equal incidence. Thus we need to multiply our probability by a further factor of 6, yielding a final probability of 0.01005, or 1/100. Thus, we should statistically expect several such coincidences in tombs already excavated.
It may be objected that I have not taken into account the coincidence of this Yeshua being a "son of Joseph". The probability of someone from this period being named Yeshua bar Yehosef is only 1/190. If we take this as our starting point, however, the presence of another ossuary identified simply as "Joseph" becomes superfluous, since the identification of the father has already been made in the Yeshua ossuary. Had there been no "Joseph" ossuary, the investigators would almost certainly have considered the family tomb to be a complete match nonetheless, and with good reason, since all of the same identifying information exists. Thus, the "Joseph" ossuary adds nothing to the strength of the match. We are left with the joint probability of the names Yeshua bar Yehosef, Maryam, and Maryam, which equals: 1/190 x 1/4 x 1/4 = 1/3040, or 0.0003289.
However, now we are choosing 3 ossuaries out of 6 to get our desired matches, and combinatorics tells us that there are 6!/3!(6-3)! = 20 possible combinations. Multiplying by 20, our new probability is 0.006578, or 1/152. This is actually a higher probability of coincidence than using the other method, before adjusting for the gender factor.
Our latent gender factor, since we are now selecting only three ossuaries, is 1/8. The actual probability of selecting 1 male and 2 females is 3/8, so we need to multiply by a factor of 3, yielding a final probability of 0.0197, or 1/50.67.
Thus, with either method, we end up with a probability that is high enough for us to expect several such name coincidences out of the hundreds of excavated tombs. In fact there are already three or four incidences of the name "Yeshua bar Yehosef", while the names "Maryam" and "Yose" are so incredibly common that we should not be surprised that they more than occasionally appear in combination. Our result of a minimum probability of 1/100 means that we should expect such a coincidence with 95% certainty after the excavation of about 300 tombs. Much more than this number has already in fact been excavated.
Our result is much more consistent with the archaeological consensus that the names on the Talpiot tomb are among the most common for that period, and thus have no probative value for identification purposes. The wildly counterintuitive claim that we can identify the subjects with 600:1 odds on the basis of four first names that were as common as Joe, John, Robert, and Edward arises from a careless choice of assumptions that ignores all the other possible combinations that would have been recognized as matches. This sort of failing is quite common in probabilistic exercises to determine coincidences, so the reader should take care not to accept such computations uncritically, particularly when they produce counterintuitive results.
© 2007 Daniel J. Castellano. All rights reserved.