Quote from: DomenicoRomano on August 09, 2013, 03:29:51 PM

Does the martingale sequence give you a better chance of winning than a single bet?

I just ran a Monte Carlo simulation.

starting bank = 1000
firstBet = 1
probability of win, each single bet = 0.5
house edge = 0.01
Amount to win = 1.01 * starting bank

Martingale rules:
Bet firstBet
If win bet firstBet
If lose bet enough to win back previous losses.
If balance = 0 then stop
If balance = Amount to win then stop
bet again

This sequence will continue until you have turned 1000 bitcoins into either > 1010 bitcoin or 0 bitcoins.

I ran this 1 million times and got the following results:

Number of wins (>1010 bitcoins) = 988311
Number of loses (0 bitcoins) = 11689
Probability of a win = 0.988311
Mean return of all sequences = 998.627
The mean of all the wins = 1010.43799

If you bet 1000 bitcoins in a single bet at probability of 0.979773138 you would also win 1010.43799, if you win.

So: With the martingale you have a 0.988311000 chance of winning.
With the single bet you have a 0.979773138 chance of winning.

Martingale wins by a p = 0.008537862.

Important to note: the mean of all the sequence is less than the amount wagered. If you do this over and over again you will be lose.

So the martingale has a better probability for turning 1000 into 1010, but how does the math change if you were to try to go from 1000 to 2000? I wonder if the martingale would still have a higher chance of winning than a single bet?