expanding e^x
Tex, here's an interesting mathematical fact:
e^x = 1 + x^2/2! + x^3/3! + x^4/4! + ...
to infinity.
So let's say you want to figure out e^1. Ok, you can start adding up terms in the sequence to get closer and closer to the answer. First, you might figure out the first term:
1
that was easy. You're also still pretty far away from the answer. So you start trying to figure out the 2nd term:
x^2/2! = 1^2/2 = 1/2
ok, not as easy, but still not hard to calculate. Now we add those terms together and get 1.5. That's a lot closer to the answer, but we're not there yet. Now we add the 3rd term:
x^3/3! = 1^3/3*2 = 1/6
hmmm, that was more tricky, but we got the answer again. Adding to our running total...
1 + 1/2 + 1/6 = 1.6666...
ok, now we're even closer to the right answer, but still far away. So we tackle the next term
1/24
and then add in the next term
1/120
and the next
1/720
forever.
Eventually, we'll get to the right answer, which is about 2.72.
Now, you may notice that as we explore ever deeper terms in the sequence, they make less and less difference in the answer. If we want to figure out e^1, the place to start is with the high order terms, the ones at the beginning of the sequence. It makes no sense to look at the 630th term in the sequence unless we have the first 629 calculated, because knowing that term only gets us one trillionth of a part closer to the answer. For all practical purposes it makes no difference at all.
Do you see what I'm getting at? This question is like the 630th term in the calculation that determines your EV in poker, but we're all still working on term 22. Handling the case where you have the nuts in a big multiway pot only to run into a river that produces a possible better hand and you're getting huge multiway action on the river and you're trying to decide whether or not to cap or go for the overcalls.... it doesn't matter what you do. Figuring out the answer to this question only makes you the tiniest little iota better, it's so insignificant that the time lost thinking about it is worth more then any EV gained through understanding.
Here are a few things that you might think about instead:
- what percentage of hands must the CO raise before it's correct to 3-bet KQ on the button?
- If the button is known to bet every street when checked to no matter what he holds, can you defend with any 2? If so, how often can he check behind some big street before it becomes incorrect?
- What should the average raise pf % be for the table before it becomes correct to limp reraise w/ aces?
I'm sure you can come up with hundreds more if you put your mind to it.
good luck.
Eric
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