Sheer Math
I'm a bit math challenged, I do admit. However I must say I was bit impressed with myself recently viewing my old college transcripts to see that I got B's in a couple of calculus classes. Wow. If I remember right that was mostly because I had had the same courses in high school getting C's. Second time was easier. My arithmetic was great ... I felt almost like there was a scaffolding in my mind where the numbers would jump around and add themselves up almost by itself. Interestingly this probably came from a game we played as boys; hand held calculators had just come out, and one of us five boys would give a long string of numbers, adding and subtracting them, figuring it up on the calculator, while the other four did it in our heads seeing who would get it right.
But the higher maths wasn't happening with me. Decades later, I was in a swimming pool with a math professor, and I asked her: "What is calculus anyway?" If I remember right she said, "For figuring the area under the curve." "Oh," I said, eruditely.
Fast forward to my seminal gambling career. Two math anomalies:
1: A 2-point conversion is about a 40% proposition, ranging from upper 30's to upper 40's in various seasons and levels of play. For 30 years on 2-point tries at the end of games which affected covers, I was on the wrong side of every one, except one. I started gathering witnesses for the streak with phone calls to other punters telling how the upcoming play would surely go. Right, right, right, right, right ... etc. One time the play went my way. It was the Ohio State/Wisconsin game, Clarett title year. OSU scored in the 4th quarter to go up 19-14, as a 6 pt. favorite. For my lungs I needed the dog. They missed it; a wild throw way over everybody's head out of the end zone. I just watched the play for the first time ever last night and I still can't believe I won that game. So the idea is: what kind of anomaly is this, going like 1-99 in 100 or so trials. How many standard deviations from the expectation is that, and is it randomly feasible? As I said I didn't see this play at the time, so therefore every single one I did see went against the result I needed. Jeez, in writing this I only now realized that the 40% thing doesn't even matter, because it wasn't about whether they made it or not, but about how whether they made it or not matched up with what I needed. Anyway, I've always had it in my head that this is an utterly impossible statistical result. Not sure if that's legit.
2. PLO. For over 30 years, when starting with a 2-pair hand, I failed to ever flop a set to this 3.5/1 shot. Well over the last 50 of these I showed to other players to verify/establish. Never. 0-for. Approximately 200 trials. How many standard deviations from the expectation is that, and is it a feasible possibility under normal concepts of randomness?
9 Replies
Why does dividing by 303 always yield a repeating fraction??
It does?
For non-zero whole numbers < 303 ??
Because 1/303 repeats and any integer multiple of a repeating decimal either repeats or ends in .9 repeating, in which case it can also be written as the next decimal value before the 9s (e.g. .9 repeating = 1, .09 repeating = 0.1, etc.)?
It can probably be easily shown that 1/(prime power) repeats for all primes not 2 and 5, and any algebraic combination (rational multiples, sums of rational multiples) also repeat.
So, basically, p/q repeats in base 10 when q has a prime other than 2 or 5 in its factorisation, I think. And p,q co-prime obvs: 606/303 doesn't count.
This can be generalised to any base. The base b expansion of a rational number in lowest terms repeats when the denominator has a prime in its factorisation which b does not.
A particular case of this is that in binary, all rational numbers that are not integers or quotients of a power of 2 (or their sums) repeat.
Now you have it.
