A true story about large numbers, no keto content but maybe interesting


#1

Like large numbers? card tricks? Stories about me in college?

When I was a physics student at Georgia Tech, my quantum mechanics professor (the late Dr. Helmut Biritz) would do something on tests that he actually thought was helpful. Some of us disagreed. I recall one question. “Calculate the angular momentum of the earth (no credit if off by more than 10^2). OK, we could get credit as long as our answers weren’t off by more than a factor of 100! I can guess a lot of things within a factor of 100, but the earth’s angular momentum isn’t one of them. He was the man that pointed out to me in 1974 that a year was approximately π*10^7 seconds. From this I concluded that π seconds was a nanocentury. The first time I had coined a unit of measure[1].

I did magic tricks as a student. Professor Biritz noted that when I had chosen a specific card out of the deck that the probability was 1/52nd even if I wasn’t using any sort of trick. He then proposed to do a trick for us no one had ever done or could ever repeat. He shuffled the deck about five times and claimed that no one had ever arranged the cards in that order and that if every human shuffled cards constantly for the next thousand years that no one could accidentally achieve the same order.

We were intrigued and went to our chalkboard. We quickly realized that this is the same problem as someone guessing all 52 cards in a random deck in order. So, if we start with a random deck, the odds of ‘guessing’ the first card are 1/52 or about 1.923%. The odds of then guessing the next one is 1/51. The combined odds of guessing them both is (1/52)(1/51) or one chance in 2652. Five cards in a row is (1/52)(1/51)(1/50)(1/49)*(1/48) or one chance in 311,875,200. This is hard to do but it is only twice as hard as winning the POWERBALL lottery.

As the deck gets smaller, the odds of picking the next card improve slightly but the cumulative probability of getting them all right rapidly becomes mind boggling. By the time you have picked eleven cards correctly, you have overcome odds of one in more than 10^18. The number of grains of sand on all the beaches of the world is about 10^18. By the time you have picked fourteen cards correctly, the odds you have overcome are more than one in 10^23. The number of all the stars in the universe is in that range…10^23. And you have just picked a little more than one fourth of the cards.

Keep going and by the time you have picked the 50th card things get easier. The next card is 50/50 and the 52nd card should be a slam dunk if you have been keeping track. But the odds of doing this are the same as two random series of shuffles resulting in the same sequence. One chance in 8*10^67. Turns out that 10^66 has a name. Unvingtillion. One chance in 80 unvingtillion of guessing them all.

What about my Professor’s claim? We have just over 7 billion people. Let’s say they can shuffle a deck once per second. One thousand years is π10^10 seconds. So, 710^9 * π10^10 is about 2.210^20. Not even close.

The current estimate for the age of the universe is 13.7 billion years (13.7 million millennia). So if the earth’s entire population had been shuffling those decks since the big bang, we would only have come up with 3*10^27 shuffles. Nowhere close enough to accomplish the task. You might argue that any two shuffles could come up with the same deck and that is theoretically true. But the odds of doing it are “astronomical” indeed.

By the way, I do this trick. But I use ‘magic’.

[1] Don’t email me pointing out anybody who came up with that before me. I do not doubt someone did, I just don’t want to hear about it.


(Bacon is a many-splendoured thing) #2

Interesting that he involved pi to calculate the number of seconds in a year. Multiplying it out, we find that the result involving pi seems to be short by a hundred thousand seconds, though your professor did say his number was approximate. Of course, total accuracy is impossible, given that they need to insert leap-seconds periodically to keep the clocks on target.

I loved the explanation of the odds of guessing a deck of cards in order. Factorials get very large, very quickly, don’t they?

As for shuffling, I read somewhere that croupiers are taught not to shuffle a deck too many times, since the deck starts to become more ordered again after a surprisingly small number of shuffles. Of course, this is particularly noticeable in the case of a deck that started in the manufacturer’s order and when the shuffles are all perfect. I’d have said intuitively that a perfect shuffle would increase the disorder of a deck better than an imperfect shuffle, but apparently I’d be wrong. I’m glad to know that there is some value in being imperfect, lol!