## Sunday, July 4, 2021

### The Probability of the Goral

In pasuk 26:54, the Torah explains how the land was divided amongst the tribes. Rashi explains exactly how the lots were picked to determine each tribe's portion.

Even though the portions were Divinely predetermined, a lot-drawing process was used to assign each tribe their portion. Rashi explains that one drum was filled with 24 pieces of paper. On 12 pieces of paper were written the names of the 12 tribes. On the other 12 were written the 12 portions that were to be assigned to the tribes. Each Nasi approached the drum and picked out two pieces of paper. One paper had the name of his tribe written on it and the other the prescribed portion of land in Eretz Yisrael. The purpose of this exercise was to prove the Divinity of division plan that allotted each tribe its portion and appease any tribe who felt it might be unfair. As such, I believe that a miracle of this type may be more greatly appreciated if we knew exactly how unlikely it would have been to happen naturally.

Suppose we had a prescribed list of which portion was to be assigned to which tribe. What would be the odds of each Nasi picking out both the name of his tribe and also the corresponding piece of land that had already been prescribed? Let us start with the first Nasi. He has 24 pieces of paper to choose from. He must pick two specific pieces of paper out of the drum. The odds of taking the first one correctly would be 1/24 and then the odds of taking the second correctly would be 1/23. However, since the two papers were taken together, the order does not matter. The rules of probability theory state that if the order of the choices is not relevant, than the odds must be multiplied by the number of possible sequences which, in this case, is two. So the odds of the first Nasi picking the right pieces is 2/(24 x 23). With 22 pieces of paper remaining, the odds of the second Nasi picking correctly will be 2/(22 x 21). And so on. The last Nasi's odds will be 2/(2 x 1) which is 1. That means that he will definitely pick the right ones. That is understandable. By the rules of probability theory, in order to find the odds of all the Nesiim picking correctly, we must multiply each Nasi's odds together. Thus, the odds may be generalized as
 212 (24 x 23 x 22 x 21...x 2 x 1)
24 x 23 x 22 x 21....x 2 x 1 is referred to as 24 factorial and expressed as 24! Thus the final expression is 212/24!. When all is totalled, the odds of the draw falling out exactly as planned without any Divine intervention would have been one in 151,476,000,000,000,000,000.

This calculation is based on Rashi's explanation in the chumash according to the Midrash Tanchuma. However, Rashi (actually written by Rashbam, Rashi's grandson,) in the gemara (Bava Basra 122a) states clearly that two drums were used. This will alter the calculation somewhat. The first Nasi would have a 1/12 chance of picking his tribe's name from the tribe drum and a 1/12 chance of picking the correct portion from the portion drum. The fact that the order does not matter will not affect the odds in this case because the choices are made from two separate groups. The odds for both drums are multiplied and thus, the first Nasi's odds will be 1/122. The odds of the second Nasi will be 1/112. And so on. The total odds will be:

 1 122 x 112 x 102 x 92 x 82 x 72 x 62 x 52 x 42 x 32 x 22 x 12

This can, in fact, be simplified as 1/12!2. The final odds will be one in 229,442,532,802,560,000. This is approximately 660 times more probable than the odds according to Rashi on the chumash.

Just to get an idea of the extent of this improbability, the odds of winning the Powerball Jackpot are approximately one in 175 million. That is over 1.3 billion times more likely to happen than this, according to Rashi on the gemara, and over 860 billion times more likely according to Rashi on the chumash. The odds of getting fatally hit by lightning in a given year are approximately 1 in 2.4 million. In fact, it is more likely for one to win the Powerball twice in one week or get fatally hit by lightning two-and-a-half times in one year than for the goral's results to have been produced naturally. This is a veritable testimony to the extent of the miracle that occurred and the Divinity of the apportioning of Eretz Yisroel to the twelve tribes.

תשע"ד: A similar drawing came up this week in דף יומי on :תענית כ"ז regarding the גורל used to determine the order of the משמרות in the בית שני. According to רש"י's understanding, there was a similar miracle at play. The probability associated with that drawing is discussed here. 1. Nice try, but your math is wrong. Each Nasi's draw is not an independent event, but a conditional event.
To give you an example of your flawed reasoning: suppose you had 3 balls in a jar labeled 1, 2, and 3, and you wanted to know what the probability was of persons 1, 2 and 3 selecting the ball that corresponded to the correct person (Person 1 picking "1", then person 2 picking "2", then person 3 picking "3", IN EXACT ORDER). There are only four possible outcomes: 1,2,3 ; 1,3; 2; and 3 - since once a ball is picked incorrectly, we stop. Thus the probability is 1/4. According to your math, the probability would be 1/6, which is wrong.
Put another way, what is the probability of rolling a die twice and getting a 1, and then another 1. This is NOT the same as the probability of rolling two dice and getting two 1's, which is 1/36!! These two events are not independent; the probability is 1/11.

2. I have to respectfully disagree. I believe yours is the math that is flawed. The fact that you choose to halt the trial when you see that the outcome will not be "true" should have no bearing on the probability.

Look at it this way: Suppose those three people in the ball example took the balls but didn't look at them and afterwards, we look to see if they chose the correct ones. Surely, there you will agree the probability is 1/6. Are you suggesting that simply changing the manner in which you conduct the same test will affect the probability? I think that's rather preposterous.

Anyway, don't take my word for it. The omnipotent Wikipedia seems to agree. The first example they give about rolling a die twice seems to be in direct contradiction to what you wrote above.

3. I'm on Shtikler's side. The reasonable assumption is that when you stick your hand into a hat (or jar, or whatever) to pick one of a collection of similar items, it is equally likely to pick each one. So if there are three balls in a jar, we assume a 1/3 probability of picking each ball. Any probability calculation should be based on this starting point. While Anonymous has described four distinct outcomes for a certain experiment, he has not provided any reasonable explanation as to why they should be considered equally likely. And Anonymous has also not explained his claims that the various sets of events are not independent. In fact they are.

4. Actually, I think that the "Rashi" commentary on Bava Batra is actually written by his grandson Rashbam. So it appears from the Bach's comments. This has the advantage that Rashi is no longer disagreeing with himself in this case.