Astounding Lottery Odds

The Torah specifies that the Land of Israel be divided among the tribes through a process of drawing lots. Rashi explains that the names of the tribes were written on 12 lots, and the boundaries of each of the 12 portions of the land were written on 12 other lots, and all 24 lots were mixed up and placed in a box. The lottery itself was conducted through Divine Inspiration by Elazar and the princes of each of the tribes. Elazar wore the Priestly garments, including the Urim V’Tumim, and announced that if the name of a certain tribe was drawn, these would be the borders of the corresponding portion of land.

The leader of the tribe mentioned by Elazar would then select two pieces from the box, and in each case, one that he selected contained the name of his tribe, and the other paper contained the region specified by Elazar. The reason for apportioning the land in this manner was to prevent discord among the tribes by making it clear that the resulting division of land was Hashem’s will. In order to understand just how convincing and persuasive this miracle was, it would be helpful to appreciate just how unlikely it was to have happened by chance.

In order to calculate the likelihood of this outcome occurring naturally, we must determine the odds of each tribal leader selecting both the name of his tribe and also the portion of land that was specified by Elazar. The first tribal prince who was called chose two lots from among the 24 that were contained in the box. The probability of correctly selecting the first is 1/24, and the odds of accurately taking the second one are 1/23.

However, because there was no requirement to select the name of the tribe first and the portion of land afterward, the order in which he chose the two pieces was irrelevant. Since Elazar’s prediction was fulfilled whether he first chose the name of his tribe and then the corresponding section of land or vice-versa, the odds of the first tribal leader successfully choosing the correct pieces is 2/(24*23).

When the second tribal prince came forward for his turn, there were only 22 pieces remaining in the box, so the probability of him correctly taking the two pieces predicted by Elazar is 2/(22*21). This will continue until the last leader, whose odds will be 2/(2*1), which is 1, since the only two remaining pieces in the box belong to his tribe, and his success is therefore guaranteed.

In order to calculate the likelihood of all 12 tribal leaders choosing correctly, we must multiply each individual leader’s odds together, which yields a probability of success of 212/(24*23*22*21*…*2*1). The denominator can also be more succinctly written as 24!, so the odds of success can be expressed as 212 / 24!, which when calculated reveals that the odds of this lottery system working out exactly as Elazar predicted by random chance is approximately 1 in 151,476,000,000,000,000,000.

This calculation is based on the explanation given by Rashi in his commentary on Chumash. However, his grandson, the Rashbam, writes in his commentary on the Gemara (Bava Basra 122a) that two boxes were used. One box contained 12 pieces on which were written the names of the tribes, and the other box contained 12 lots on which were written the boundaries of the respective portions of the land. When it was his turn, each tribal prince selected one from each of the two boxes.

This explanation alters the calculation of the likelihood of random success, as the first leader now has a 1/12 probability of choosing the name of his tribe from the first box, and a 1/12 probability of selecting the area of land predicted by Elazar. As a result, the odds of success for the first tribal prince are 1/122, while the probability of the second leader choosing correctly are 1/112, which is continued all the way until the final leader, whose odds are 1/112, which is 1, since his success is guaranteed. Multiplying the odds of all of the leaders together, the likelihood of Elazar’s predictions being randomly fulfilled in this model is 1/(122*112*102…*22*12). This can also be expressed as 1/(12! 2), and when calculated yields a probability of random success of 1 in 229,442,532,802,560,000, which is approximately 660 times more likely than the odds according to the explanation given by Rashi in his commentary on Chumash.

To appreciate the sheer magnitude of how improbable this would be, the odds of winning $1,000,000 in the New York Mega Millions lottery are approximately 1 in 18 million, which is more than 12 billion times more likely than the successful fulfillment of all of Elazar’s predictions according to the Rashbam’s commentary on the Gemara, and a whopping eight trillion times more probable according to Rashi’s commentary on Chumash. It would actually be more likely to win the lottery twice in one week than for the tribal lottery to occur according to random chance, which gives us a newfound appreciation of the extent of the miracle that Hashem performed in dividing Eretz Yisrael among the tribes.

Q: Which book of Tanach was co-authored by Pinchas?

A: The Gemara teaches that Yehoshua authored the book of Yehoshua until the verse that records his death (Yehoshua 24:29), at which point Elazar became the author until the verse recording his death (Yehoshua 24:33), and the remainder of the book was completed by Pinchas.


Originally from Kansas City, Rabbi Ozer Alport graduated from Harvard, learned in Mir Yerushalayim for five years, and now lives in Brooklyn, where he learns in Yeshivas Beis Yosef, is the author of the recently-published sefer Parsha Potpourri, and gives weekly shiurim. To send comments to the author or to receive his Divrei Torah weekly, please email oalport@Hamodia.com.