martingale math - what say ye?

Quote: MathExtremist

For the craps pass bet E = 1 - 7/495 = 0.98586.

I worked out the expectation for the pass line bet:

(8/36) - (4/36) + 2(1/36) - 2(2/36) + 2(8/180) – 2(12/180) + 2(25/396) – 2(30/396) = - .0141

I see 7/495 = .0141, but how did you get 7/495?

By hand, without using Excel. It's the same value.

Here's how Wolfram Alpha solves "(8/36) - (4/36) + 2(1/36) - 2(2/36) + 2(8/180) – 2(12/180) + 2(25/396) – 2(30/396)"

Thanks.

Quote: P90

No, it's not mathematically sound. It's only neutral (not advantageous) for a fair bet.

In casino games, martingale increases the effective house edge. I'll quote one of my older posts on this.

...
For n-deep martingale in roulette, the possible outcomes of a martingale progression are:
1-(20/38)^n: Win 1 unit
(20/38)^n: Lose 2^n-1 units
For a total EV of 1-(20/38)^n-(2^n-1)(20/38)^n = 1-(20/38)^n*2^n = 1-(40/38)^n ~= 1-1.0526^n

Or, for other games, 1-(1+HA)^n.
It is a very simple function. For any HA>0, it exponentially approaches -infinity as possible martingale progression depth (n) increases.
If you just martingale 2 bets deep, you are playing a 10.8% house edge game instead of 5.26%. If you do it 3 bets deep, 16.7%. Playing $10-$1,000 game will let you martingale 7 bets deep, for a house edge of 43%.

Okay,

So again, I don't agree with this. If I bet (call it whatever system you want) so that my next win covers ALL my loses, including the VIG, then each bet only has a house advantage of 5.26%...it doesn't compound as you say because you cover that loss with our next bet.

Here is another excel spreadsheet that shows this nicely using a lay bet and 5% VIG. Both the roulette and the lay bets show a profit over a million rolls. The lay bet showed a $597,000 profit. I'm not sure where you guys are getting that a martingale system isn't mathematically sound. It isn't practical, but that wasn't the question.

https://docs.google.com/open?id=0B3eruTU5Xu_jOTg0YjgxMGMtMTk5Yi00NzllLWEzZGUtMGZiNzE0OTE2YTNm

Each individual bet has indeed the house advantage of 5.26% for roulette.

But each string of bets in a x-level Martingale is only ever going to win you 1 unit. So what P90 (and I) did was look at the expected value of that string. Either you win 1 unit of lose X^2 units. Whats the expected value? And it's not 5.26% for the complete string.

Your sheet for roulette is 40,134 spins long with one 15 long losing streak.... something you expect about once in 40,000 spins. As you used RAND, I've seen streaks up to 18 in there. In short, for the bank roll you've given yourself (1 million) and the max number of spins you wish to risk, I'm not surprised you've never gone bust. I don't see a million results here, but I suspect that might the number of items I seen.

This doesn't make a Martingale "Mathematically sound". You've shown only that it works in some experiments. Try the experiments with a "cap" on your bankroll (or amount willing to bet on a spin). I think you'll see more failures to show a profit.

Quote: thecesspit

Each individual bet has indeed the house advantage of 5.26% for roulette.

But each string of bets in a x-level Martingale is only ever going to win you 1 unit. So what P90 (and I) did was look at the expected value of that string. Either you win 1 unit of lose X^2 units. Whats the expected value? And it's not 5.26% for the complete string.

Your sheet for roulette is 40,134 spins long with one 15 long losing streak.... something you expect about once in 40,000 spins. As you used RAND, I've seen streaks up to 18 in there. In short, for the bank roll you've given yourself (1 million) and the max number of spins you wish to risk, I'm not surprised you've never gone bust. I don't see a million results here, but I suspect that might the number of items I seen.

This doesn't make a Martingale "Mathematically sound". You've shown only that it works in some experiments. Try the experiments with a "cap" on your bankroll (or amount willing to bet on a spin). I think you'll see more failures to show a profit.

ONly 40,000 - you are right, but I refreshed 25 times and kept track of the results. I'm not excel savy, but I am sure there is a way to tell the thing to run the sheet x number of times, and sum the results. But whatever.....

Like I said, it isn't practical.

But....

Perhaps I am missing something. For something to be mathematically sound in your mind, what do you think that means?

Proof by induction or deduction, rather than an example or simulation.

I've seen two different proofs that the Martingale does or does not work with infinite bank rolls and doubling. The biggest problem is the word infinity... things go wibbly at infinities.

I have no problem with simulation, as it uncovers interesting (*) behaviour's as you have shown. I did similar things with various progressions to see what they looked like. As I've said, have a play with your sums with a practical values of streaks.

(*) Interesting in the sense of discovering new stuff for yourself.