Tuesday, July 8, 2014

:כ"ז Similar Raffles

וחלקום והעמידום על עשרים וארבעה בללום ונתנום בקלפי בא ידעיה ונטל חלקו וחלק חבריו שש בא [חרים] ונטל חלקו וחלק חבריו שש וכן פשחור וכן אימר

How fitting that this גמרא falls out in פרשת פינחס. In an installment of על פי חשבון, I attempted to calculate the probability of the Divinely "rigged" גורל falling out exactly like it did. So, then, what would be the odds of this one coming out like it did without the obvious Divine intervention?

I have to admit, I'm not sure I fully understand the exact process and exactly who picked what. I also have not investigated the מפרשים sufficiently. So I will keep it as simple as possible. Therefore, we will assume that the one picked for each משמר did not have to pick the name of the משמר first. In other words, ידעיה picked first. He came up with 6 names, all of which belonged to the ידעיה family but one was actually named ידעיה. Let us assume that he picked all 6 at the same time*. Using simple nCr notation, there would be 24C6 different possible combinations for ידעיה. For the uninitiated, that is simply a fanyc way of expressing the following: 24! / (24-6)! x 6! Simply put, the different possible combinations ar
24 x 23 x 22 x 21 x 20 x 19
6 x 5 x 4 x 3 x 2 x 1
The reason we have to divide by 6 x 5 x 4...  is because order doesn't matter, so there are that many different combination which are essentially one in the same.

I probably have not sufficiently explained the above concept for those who are not familiar but if we move on, I believe the probability can be expressed thusly:
1
24C6 x 18C6 x 12C6
The probability of the final group is essentially 1 since there are only those 6 tickets left so we really only need examine the probability of the first three groups. That computes to
1
134596 x 18564 x 924
Which is 1 in 2,308,743,493,056. Although both this and the גורל of פינחס both involved 24 tickets, since this one was drawn in groups of 6, the probability is in fact nearly 100,000 greater than that of פינחס. Nevertheless, as was discussed in my parallel calculation of פינחס, the odds would still be close those of getting fatally hit by lightning twice in one year! Again, this is a veritable testimony to the extent of the miracle that occurred and the Divinity of the raffle.

*Well I just looked a little deeper into this.:רש"י ערכין י"ב explains that the raffle happened exactly the way I just assumed that it didn't. So first the head of the משמר picked his own name and then the other 5. This complicated things slightly. The probability would therefore be:
1
24 x 23C5 x 18 x 17C5 x 12 x 11C5

Which - let's just skip to the end - is 1 in 498,688,594,500,096, or over 200 times less probable than originally thought.

You want to simplify all this? According to רש"ש, there was nothing particularly miraculous going on here at all. There were in fact 4 separate raffles of 6 whereby each of the 4 משמרות would draw to determine the order of their respective subdivisions.




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