Here is what I'm thinking, and it has to do with the Gambler's Fallacy and Law of Large Number. The Gambler's Fallacy states that due to the probability of an event is statistically independent and has no memory of previous outcome, regardless how unlikely the previous outcome has been. I saw some of the explanations for Gambler's Fallacy are that it's due to small sample size. But my understanding is that it has nothing to do with sample size. If a gambler were to bet on the toss of a fair coin, his chance would be 50/50, regardless if the previous result was a streak of 20 tails, or 200 tails, or 20^20 tails, however unlikely such event might be.
To expound on this idea further, picture a cosmic gambler who has been tossing a fair coin since the beginning of the universe, and who has been getting tails all these time. His chance of getting a head instead of tail is still 50/50. Which means, he could been getting tails until the end of the universe. Because to state otherwise would be violating the Gambler's Fallacy.
However, this would be an apparent violation of the Law of Large Number, which states the average of the results from a large sample should be close to the expected value. In other words, for our cosmic gambler, for all his coin tosses, the ratio of head and tails should be exactly 50/50 by now(13 some billions years).
A rebuttal could be made at the high unlikeliness of getting all tails on a fair coin toss for 13 billion years, let alone an eternity. Unlikely, perhaps, for our normal living. But as Sherlock would remind us, improbable is not impossible. So let's now imagine instead of one single cosmic gambler, we have an infinite number of cosmic gamblers who have been tossing fair coins since the beginning of the universe. It's abstract math we're talking here, so bear with me. Now using the Infinite Monkey Theorem, which states that for infinite number of monkeys typing on keyboards, given enough time, they could produce a Shakespeare. That would imply, given an infinite number of gamblers, there would be at least one that would end up never getting a head on the tosses. In fact, there would be an infinite number of gamblers getting that kind of result.
Similarly, we can prove that there could be gamblers who would get exactly one head out of eons of tosses. And then gamblers who would get two heads, and three heads, and so on.
And similarly, we can show there would be gamblers who would get only heads, and only one tail, and only two tails, and so on.
As it happens, if you add all the coin tosses from these timeless gamblers together, the overall probability should be very close to ½, thereby preserving the Law of Large Number.
Let's assign these gamblers to groups based on their coin toss results. The ones getting all tails are PH0, and the ones getting only one head PH1, then PH2, and so on. Given there are infinite numbers of gamblers, each group would have infinite members. However, in counting, some infinite numbers would be greater than others. Clearly, the groups that contain gamblers getting even coin toss result would be far greater than the marginal groups such as PH0, PH1.
If we assign getting head on a toss is a favorable result and getting tails is not, then we can demonstrate that the gamblers in PH0 are as down on their luck as they can be and their brethrens in PH1 and PH2 are not much better off. And conversely, their counterparts who are getting all heads are some lucky bastards. And given these lucky and unlucky groups contains gamblers much much less than the ordinary groups, it fits our normal perception of what luck is, which is an highly improbable event.
If all the above is true, then it should show that luck can be proven mathematically. Any thoughts?