### 2.27.2007

## Probability

I had a quick look at Andrey Feuerverger's calculations of the probability of the Talpiyot tomb being Jesus's tomb.

First of all, it is extremely important to remember that in the real world, the question "what is the probability of X" is almost always meaningless. Conditional probabilities are the only probabilities that can be calculated, i.e. "what is the probability of X, given that we already know Y".

Feuerverger's method seems to be to calculate the probability of finding the name "Jesus son of Joseph", "Mariamne", "Maria" and "Yose" in the same tomb and obtains a 1/600,000 chance of this. Given that there are about 1000 known tombs of 1st century families in Jerusalem, he concludes that the expected number of tombs with these names is 1/600 (even arbitrarily throwing in a fudge factor of 4 to obtain a more conservative estimate). Given the existence of the tomb we have here, he concludes that that the chance that it belongs to Jesus is 1 / (1 + 1/600), which is 600/601, i.e. there is about a 1/600 chance that it isn't Jesus' tomb.

Now of course we should recognize the conditional assumptions built into this calculation, as Jay noted in Ben Witherington's blog -- we are assuming that we know independently that (1) Jesus's northern Israel-based family would actually be buried in a family plot in Jerusalem, in the southern part of the country, (2) Jesus' tomb would contain a "Mariamne" (and a "Judah" and "Matia", but not have boxes with his other relatives' names on them), (3) Jesus would be identified as "Jesus son of Joseph" (last I checked, he was usually referred to as "Jesus son of Mary" if he was identified as someone's son at all).

Even granting these assumptions, Dr. Feuerverger's calculations still seem to have problems.

The 1/600,000 chance is obtained by taking 1/190 (P(random male is "Jesus son of Joseph)) times 1/160 (P(random female is "Mariamne")) times 1/20 (P(random male is "Joseph") times 1/4 (P(random female is "Mary")) * fudge factor of 4.

The problem here is that the correct way to obtain the probability of this cluster of names is to take P(one of five men in the tomb is "Jesus son of Joseph") * P(one of the four remaining men in the tomb is "Joseph" | P(there is already a "Jesus son of Joseph")) * P(one of two women in the tomb is "Mariamne") * P(the other woman in the tomb is "Mary").

The probability of one of five men being "Jesus son of Joseph" is closer to 5/190 (actually, it's 1 - (189/190)^5, but close enough), or about 1/40.

The probability of one of the four remaining men being "Joseph", given that we already know that there's a "Jesus son of Joseph", is...pretty darned high, as Jay notes. Now if you *assume* that this isn't Jesus's legal father, but another Joseph, then the probability of one of the four remaining men being "Joseph" is (1 - (19/20)^4), or about 1/5.4.

The probability of one of two women being "Mariamne" is (1 - (159/160)^2), or about 1/80.

The probability of the other woman being "Mary" is 1/4.

If we don't assume anything about the Joseph in the tomb, then the probability of a random tomb having these names is (1/40)*(1/80)*(1/4) = 1/12800, making it about a 1/13 chance that this isn't Jesus's tomb (again, assuming that Jesus is buried in a tomb, and that the unusual location and surprise names in the tomb can be reconciled independently with what we know about Jesus, and also discarding Dr. Feuerverger's fudge factor of 4).

If we assume that Joseph is actually not Jesus's father (which means that the presenc of anothe Joseph in Jesus's family tomb needs to be independently reconciled) then the probability of a random tomb having these names is 1/12800 * 1/5.4 = 1/69000, or about a 1/70 chance that this tomb isn't Jesus's tomb. Again, these estimates are conditional upon throwing out a lot of evidence that suggests that this isn't even Jesus's tomb in the first place.

First of all, it is extremely important to remember that in the real world, the question "what is the probability of X" is almost always meaningless. Conditional probabilities are the only probabilities that can be calculated, i.e. "what is the probability of X, given that we already know Y".

Feuerverger's method seems to be to calculate the probability of finding the name "Jesus son of Joseph", "Mariamne", "Maria" and "Yose" in the same tomb and obtains a 1/600,000 chance of this. Given that there are about 1000 known tombs of 1st century families in Jerusalem, he concludes that the expected number of tombs with these names is 1/600 (even arbitrarily throwing in a fudge factor of 4 to obtain a more conservative estimate). Given the existence of the tomb we have here, he concludes that that the chance that it belongs to Jesus is 1 / (1 + 1/600), which is 600/601, i.e. there is about a 1/600 chance that it isn't Jesus' tomb.

Now of course we should recognize the conditional assumptions built into this calculation, as Jay noted in Ben Witherington's blog -- we are assuming that we know independently that (1) Jesus's northern Israel-based family would actually be buried in a family plot in Jerusalem, in the southern part of the country, (2) Jesus' tomb would contain a "Mariamne" (and a "Judah" and "Matia", but not have boxes with his other relatives' names on them), (3) Jesus would be identified as "Jesus son of Joseph" (last I checked, he was usually referred to as "Jesus son of Mary" if he was identified as someone's son at all).

Even granting these assumptions, Dr. Feuerverger's calculations still seem to have problems.

The 1/600,000 chance is obtained by taking 1/190 (P(random male is "Jesus son of Joseph)) times 1/160 (P(random female is "Mariamne")) times 1/20 (P(random male is "Joseph") times 1/4 (P(random female is "Mary")) * fudge factor of 4.

The problem here is that the correct way to obtain the probability of this cluster of names is to take P(one of five men in the tomb is "Jesus son of Joseph") * P(one of the four remaining men in the tomb is "Joseph" | P(there is already a "Jesus son of Joseph")) * P(one of two women in the tomb is "Mariamne") * P(the other woman in the tomb is "Mary").

The probability of one of five men being "Jesus son of Joseph" is closer to 5/190 (actually, it's 1 - (189/190)^5, but close enough), or about 1/40.

The probability of one of the four remaining men being "Joseph", given that we already know that there's a "Jesus son of Joseph", is...pretty darned high, as Jay notes. Now if you *assume* that this isn't Jesus's legal father, but another Joseph, then the probability of one of the four remaining men being "Joseph" is (1 - (19/20)^4), or about 1/5.4.

The probability of one of two women being "Mariamne" is (1 - (159/160)^2), or about 1/80.

The probability of the other woman being "Mary" is 1/4.

If we don't assume anything about the Joseph in the tomb, then the probability of a random tomb having these names is (1/40)*(1/80)*(1/4) = 1/12800, making it about a 1/13 chance that this isn't Jesus's tomb (again, assuming that Jesus is buried in a tomb, and that the unusual location and surprise names in the tomb can be reconciled independently with what we know about Jesus, and also discarding Dr. Feuerverger's fudge factor of 4).

If we assume that Joseph is actually not Jesus's father (which means that the presenc of anothe Joseph in Jesus's family tomb needs to be independently reconciled) then the probability of a random tomb having these names is 1/12800 * 1/5.4 = 1/69000, or about a 1/70 chance that this tomb isn't Jesus's tomb. Again, these estimates are conditional upon throwing out a lot of evidence that suggests that this isn't even Jesus's tomb in the first place.

### Comments:

hey eddie,

found your blog through james choung's. i see you're out in LA, i'm sure conquering every square inch with much gusto. hope all is well.

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found your blog through james choung's. i see you're out in LA, i'm sure conquering every square inch with much gusto. hope all is well.