Is Chance proveable?

There is some rationality mixed in with a lot of nonsense here.

The table limit is used to prevent the results being skewed by a player betting huge amounts. If ten players bet the same amount, the casino is assured of a profit in every 100 plays. If one player is betting $10,000 and the others are betting $10, the casino must hope that the big bettor is not lucky.

BTW: The odds in 100 plays at a dice table are as follows: 40% of the players will win; 8% will break even; 52% will lose. The casino does not want a really big bettor to be one of the 40% who win. That is the reason for the table limit. An oil sheik who wanted to double up after every loss starting with a $1000 bet might be allowed to have no table limit. The casino only risks $1000 per series of bets, and is likely to win a million when the sheik runs out of money or nerve. They know he will hit a losing streak sooner or later.

If a coin, roulette wheel, whatever is not rigged, past behavior just does not influence what happens next. As pointed out by somebody, a casino does not want people sitting at a roulette table not betting while waiting for a run of Black or Red to occur. The casino probably would not have objected if they had made a minimum bet every play and then bet a huge amount on Black after a series of Reds. If their theory had any merit, this strategy would pay in the long run.

BTW: In Las Vegas (and probably in other casino towns), you can purchase roulette monitoring forms. I have seen them in use at Vegas casinos in off hours. The user stands close enough to the table to observe, but does not take a seat. One strategy is to record the results of 380 plays. Each number is expected to come up 10 times. The gambler then sits down and bets on the numbers which have occurred less than ten times, assuming they are now more likely. Oddly enough, some bet on the numbers which have come up more often, assuming that the wheel is unbalanced in favor of those numbers. Either strategy takes a lot of patience. 380 spins is about 30 minutes to an hour. The strategy is used at off hours because there are empty seats, allowing the player an unobstructed view, and also allowing him/her to take a seat when his bookkeeping tells him it is time to bet.

The expected result of a large number of plays is a bit counterintuitive. If you flip coins a huge number of times and count heads and tails, AbsoluteValue(Heads - Tails) is expected to grow without bound, while Heads/Tails is expected to approach ½. Consider the following.

* P(even in 10 flips) = .246 094
* P(even in 100 flips) = .079 589
* P(even in 1000 flips) = .025 225
* In 1024 flips if (Heads - Tails) = one million, Heads/Tails would equal ½ to 18 digits of precision.

The more you flip a coin, the less likely an exact 50-50 match becomes. As you flip it many times, it becomes very likely that Absolute(Heads - Tails) will equal 100 or more.

Somebody made an erroneous post about the half life of a radioactive substance. The half life is the time in which half the atoms are expected to decay. If the half life is one minute, half are expected to decay in one minute; In two minutes 3/4 are expected to decay; In 3 minutes, 7/8; Et cetera. If there is only one atom of the substance, it will last one minute half the time; It will last two minutes 1/4 of the time; It will last 3 minutes 1/8 of the time; et cetera.