very nerdy and fun! Thank you
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More math …
If the probability of an event is p, and there are N=1/p opportunities where the event might occur, then as p get smaller and smaller, the probability the event will occur at least once in those N trials is about 63.3%.
More formally, this probability converges to 1-1/e.
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I’ve thought that a really cool fact too! Or to put it in another way some might find more relatable … to buy 10 lotto tickets when each has a 1/10th chance of being a winner … or 100 tickets if each has a 1/100th chance, or buying a million if each has a millionth chance … as we keep increasing that number, the chances of your large stash of tickets including at least one winner approaches the value you’re referring to: 1-1/e (or about 63%). That has interesting implications. It means that I can choose any threshold of confidence I wish to achieve, and compute a finite number of tickets I need to buy to achieve it. Do I want to reduce chances of not getting a winning ticket down to below 2%? Then just take the 1/p number (N) and quadruple it. Buy 400 lotto tickets if each has a 1/100th chance, and you have better than 98% chance of at least one winner. Or make it higher yet to get as close to certainty as you wish. This means that in any infinitely long sequence of non-patterned numbers (like pi or any other irrational number), you can be guaranteed with all the certainty you want that the entire Bible, complete works of Shakespeare, Darwin’s “Origin of Species” is included in there somewhere (keeping in mind that the number of digits you need to get such a guarantee would be beyond imagination or any simple expression in scientific notation). You have to just bank on infinity to reach for such a conceptuality. But hey … it’s “there”!
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Yes, I’ve always thought that the “almost” certainty of finding a given pattern, like the works of Shakespeare, in a random sequence of sufficiently large length, a good way to think about infinity – mind boggling.
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If we are talking about thinking probabilistically, how about the “Monty Hall problem” (spoiler alert: don’t look at the reference if you want to think about it) Monty Hall problem - Wikipedia
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
That is, is the best strategy to switch your choice, or does it not matter whether you switch or not?
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