Quote:ThatDonGuyThe fraction of wins approaches 0.5, but that does not necessarily mean that the wins minus losses value approaches zero - I have already shown a counterexample to that statement.

Here is the definition of the Law of Large Numbers from Wolfram Mathworld (emphasis mine):

A "law of large numbers" is one of several theorems expressing the idea that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero.

Yes - similar to the principle of "infinite monkeys and infinite typewriters," you "should" "eventually" catch up for any positive probability bet. It might take quadrillions of years, but it "should" happen "eventually." Of course, keep in mind that, under the same conditions, Martingale has positive EV as well.

I ran a 50/50 bet simulation using D'Alembert with the stop condition that you had to stop if you wiped out a bankroll of one billion (with a B) initial bets. At first, every sequence won, and the "per run" profit got as high as 6000 - then, about one billion runs into the simulation, it hit a bet that kept trying to crawl its way back to success but eventually hit the bankroll limit, and the per run profit (including all of the ones that led to the 6000 per run rate up to that point) was now a 6000 per run average loss. Note that the overall bet success was 49.99893%. This is in line with a general principle of pretty much every system - the fraction of wins may be high, but the ratio of average loss amount to average win amount makes up for it.

When you ran the simulation, did you see if the simulator broke the LLN? Did it go backwards, like. 49,9% to 49.8% to 49.7% to 49.3% etc. before finishing up at 49.99893%?

Quote:JyBrd0403When you ran the simulation, did you see if the simulator broke the LLN? Did it go backwards, like. 49,9% to 49.8% to 49.7% to 49.3% etc. before finishing up at 49.99893%?

How far down does the winning percentage have to go before you consider it to "break the LLN"? If you consider any drop to "break LLN," then yes, it did go backwards, and quite a lot - because if the winning percentage is under 50%, any loss will cause the percentage to drop (as you have the same number of wins, but one more total play than before). One of the reasons LLN "works" is, if the winning percentage is under 50%, it will go up if your next two plays are a win and a loss (in either order) - or, for that matter, if your next 2N plays are N wins and N losses, again in any order.

Suppose you have W wins and L losses, with W < L; your fraction of wins is W / (W + L) < 1/2.

After N more wins and N more losses, it is (W + N) / (W + L + 2N).

L > W, so multiply both sides by N: LN > WN

Add W

^{2}+ WL + WN to both sides:

W

^{2}+ WL + WN + LN > W

^{2}+ WL + 2WN

(W + N) (W + L) > W (W + L + 2N)

(W + N) / (W + L + 2N) > W / (W + L)

Your fraction of wins has gone up

However, LLN is not "set in stone"; if I play a 50/50 game a billion times and win 49.99999% of them, that does not mean that I have to win at least 49.99999% of the next billion as "otherwise that breaks LLN as my fraction of wins will go down." Believing otherwise is pretty much the definition of the Gambler's Fallacy.

Quote:ThatDonGuy

However, LLN is not "set in stone"; if I play a 50/50 game a billion times and win 49.99999% of them, that does not mean that I have to win at least 49.99999% of the next billion as "otherwise that breaks LLN as my fraction of wins will go down." Believing otherwise is pretty much the definition of the Gambler's Fallacy.

Really? To defend the simulation results you're now going to throw out the Law of Large Numbers? Really?

Hear tell it's a fundamental theorem of probabilities. Fundamental meaning if you throw that out you can throw out the rest of probability math as well.

You know when I read on the internet they usually talk about 10,000 trials for the LLN to take effect. You now want to toss out billions of trials? Really?

There is no shame falling for Gambler's Fallacy, even an incredibly smart guy like d'Alembert did.

Quote:JyBrd0403Really? To defend the simulation results you're now going to throw out the Law of Large Numbers? Really?

What do you mean by "defend the simulation"? Are you claiming that, since the percentage of wins dropped at all once it got below 50%, then it is flawed because LLN says "that can't happen"?

I will repeat my earlier question:

How far down does the winning percentage have to go before you consider it to "break the LLN"?

Here's another question for you:

Suppose, in the first billion trials, there were 500,300,000 heads - a 50.03% win rate. Then, in the next billion trials, there were only 500,100,000 heads - a 50.01% win rate. The rate dropped from 50.03% to 50.02%, "as expected by LLN."

Now, switch the two sets of trials; the rate now increases from 50.01% in the first billion to 50.02% in the first two billion. This appears to contradict LLN.

Are you saying that the two sets of one billion events weren't equally as likely to be the "first set"?

Quote:DiscreteMaths2I mean the betting system is literally named after Jean le Rond d'Alembert, the guy who thought the chance of a coin flip changed based on past results: https://www.cs.xu.edu/math/Sources/Dalembert/croix_ou_pile.pdf

There is no shame falling for Gambler's Fallacy, even an incredibly smart guy like d'Alembert did.

-Another question. Pierre plays against Paul on this condition, that if Pierre brings heads on the first toss, he will pay an ecu1

to Paul; if heads is brought only on the second toss, two ecus; if on the third toss, four, & thus in succession.-

It looks like this must have been some early work D'Alembert did on the subject, if it's legitimate at all. 1+2+4... is a martingale progression, not the D'Alembert progression.

Quote:ThatDonGuyWhat do you mean by "defend the simulation"? Are you claiming that, since the percentage of wins dropped at all once it got below 50%, then it is flawed because LLN says "that can't happen"?

I will repeat my earlier question:

How far down does the winning percentage have to go before you consider it to "break the LLN"?

Here's another question for you:

Suppose, in the first billion trials, there were 500,300,000 heads - a 50.03% win rate. Then, in the next billion trials, there were only 500,100,000 heads - a 50.01% win rate. The rate dropped from 50.03% to 50.02%, "as expected by LLN."

Now, switch the two sets of trials; the rate now increases from 50.01% in the first billion to 50.02% in the first two billion. This appears to contradict LLN.

Are you saying that the two sets of one billion events weren't equally as likely to be the "first set"?

The LLN says that the win percentage will move closer to the EV the more trials you run. Most math experts I see on the internet consider 10,000 trials enough for the LLN to kick in, meaning that after 100 trials you might be at 49%, after 500 trials you may move down to 47%, but after 10,000 trials the win percentage will have moved closer to the EV then it was at 100 trials. So, you can expect the win percentage to move closer to the EV every 10,000 trials or so.

So, yes according to the LLN in billion trial intervals, any move backwards in the win percentage from the 1st billion to the 2nd billion (and in this case a 200,000 unit move backward) would definitely be breaking the LLN. Especially, when the argument is that you would continue to move further away from 0 after this point.

Quote:DiscreteMaths2All D'alembert does is modify the frequency distribution of the net gains / net losses at the end of a gaming session, which when you take all the sessions into account approaches the EV of the game after more and more sessions. Any other betting system you can think of approaches that same EV. If me and you bet on fair coin flips, the rate and manner we choose bet amounts isn't going to make the fair coin not 50/50. I mean the betting system is literally named after Jean le Rond d'Alembert, the guy who thought the chance of a coin flip changed based on past results: /math/Sources/Dalembert/croix_ou_pile.pdf

There is no shame falling for Gambler's Fallacy, even an incredibly smart guy like d'Alembert did.