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Tisha B'Av Probability Analysis

By Morris Engelson

Hitting a target by chance proves nothing even if the probability against it is a million to one. But hitting that target multiple times after the first arrow is identified and circled, is highly significant. The “first arrow” of Tisha B'Av comes with the spies in Parshat Shelach as the people wept and G-d decreed that, “I will establish this night for them as a time of weeping throughout the generations.” I once gave a talk on this and was asked to provide a mathematical analysis of this assertion. Here it is.

The following is a list of ten significant times “of weeping” in our 9 Av calendar. Destruction of first and 2nd Temples, Betar captured with 580,000 casualties, pagan temple to Jupiter built on Temple Mount and Jerusalem renamed Aelia Capitalina, King Edward of England signs decree of expulsion in 1290, expulsion from Spain in 1492, Shabbtai Zvi born in 1626, Germany declares war on Russia on August 1, 1914 to start WWI, Reinhard Heydrich receives orders, dated July 31, 1941, to implement “final solution”, first transport for extermination reaches Treblinka on July 23, 1942.

The likelihood of a random event happening on one specific day out of 365 is one (1) in 365, or a probability of 1/365. Probabilities of multiple independent events are multiplied to yield the likelihood of several events taking place. Hence, four hits on Tisha B'Av is 1/365 multiplied by itself four times or (1/365)4 = 1/1.8 with 10 zeros, 6 events are a probability of one in 2.4 followed by 15 zeros, and ten such events have a likelihood of one out of 4.2 followed by 25 zeros.

The above numbers are very impressive, but the result is deceptive. Using the same logic, suppose we had 38 happenings on Tisha B'Av? What is the probability for that? It is one out of 2.3 followed by 97 zeros. But why stop at 38? How about 100 or more events on Tisha B'Av? I can’t give you the answer because my calculator stops at 100 zeros and we get thousands and thousands of zeros in the probability result. This is as close to zero likelihood as possible. But the implication is false. So let us look at this matter again, but from a different perspective.

Suppose there are 1000 significant events that fit a day of mourning in the last 2000 years. What is the probability that one of these will fall on Tisha B'Av? We proceed as follows: Either an event falls on Tisha B'Av, for which the probability of one event is 1/365, or it falls on some other day of the year for which the probability is 364/365. The probability that all 1000 events will fall on 364 days of the year and none on number 365, which is 9 of Av, is (364/365)raised to 1000 power = 0.0643. This means that the likelihood that one of these events will fall on Tisha B'Av is 1-0.0643 = 0.936. Two events is the square of this number which is 0.876 and four events is the square of the last number, and that is 0.767. There is a 77% probability that four events will happen on Tisha B'Av on a purely random basis. This is very, super very, far from the one chance in 1.8 followed by 10 zeros that I calculated previously. Clearly, it is not enough to deal with a calculation based on one event for one day out of 365 days in the year. We must also consider the number of possible events that are candidates for this calculation.

This means that we must establish a list of candidate events based on historical parameters before we proceed with a calculation of probabilities. I listed 10 events. Suppose the total list involves 20 candidate events, only 10 of which happened on Tisha B'Av. We then ask, what is the probability for this situation? The likelihood that any one of these will fall on Not Tisha B'Av is 364/365, and the likelihood that all 20 will fall on a day other than Tisha B'Av is 0.947. Hence the probability for one of these falling on Tisha B'Av is 1-0.947 = 0.053 and the likelihood of 10 events on Tisha B'Av is 0.053 raised to the tenth power, which is 1.7 x 10-13. This is an impressive number that is statistically very significant.

I could stop here and claim that the 10 events I listed in my talk are significant, and the odds are very impressive even if we find ten more such events that did not fall on Tisha B'Av. But let’s get more selective and more realistic in order to improve the quality of the analysis. The choice of which historical events fit depends on the choice of criteria and then on a judgment as to which events fit the criteria. I would choose as a criterion, events that are in the category of an existential threat to the Jewish people. I argue that the loss of each Temple fits my choice. I would also argue that the loss of Betar and the accompanying loss of statehood are also in this category. I would also argue that plowing over the ground of Jerusalem plus renaming the city plus the building of a temple to Jupiter plus making the teaching of Torah a capital offense fits the criteria. I would argue that one or several of the indicators respecting the Holocaust are also in this category.

The expulsion from England is not sufficient for me, nor is matters connected to Shabtai Tzvi. The expulsion from Spain is a question mark. This gives us between four and six events that fall on Tisha B'Av and one (the start of WWII, not on my list) not on Tisha B'Av. Plus I will add one more item not on Tisha B'Av that somebody might find. I don’t know what that can be, but somebody might add something that I did not list – say the crusades? This yields a probability of 0.02 for one event falling on TishaB'Av and one chance in one billion (one followed by 9 zeros) for 5 out of 7 falling on Tisha B'Av. Different criteria and different choices of events will yield different probabilities. I believe that my calculation, based on only 5 hits on Tisha B'Av out of 7 possibilities, is conservative. This, conservative, result is still only one chance out of a billion. It is difficult to argue that such odds are due to purely random chance.

Note: Probability results can be highly subjective, hence not all will agree with my analysis. For example, someone not connected to Torah might not accept that the first Temple was destroyed on 9 Av, since there is no independent historical corroboration. Likewise, historians differ as to whether the expulsion deadline from Spain was on 9 Av or on 7 Av. The “final solution” order date of July 31, 1941 is 7 Av and not 9 Av. There is evidence that Heydrich helped prepare the content and “knew” about it prior to 7 Av. But the date on the document is 7 Av, and we don’t know whether Heydrich received the document that day or later. All we can say with certainty is that it was near 9 Av.

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