Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last match, I lose my money and I will bet double next time (2.000 USD)… If they don't win again, I will bet double again (4.000 USD)… I will continue doubling my bets until Barcelona wins…
Barcelona would have to lose 10 matches in a row, so that I would run out of my finances (1 million USD)… Which is not going to happen…. I think Barcelona haven't lost 10 matches in a row in La Liga in the entire history of the competition…
If Barcelona wins the match and I win the bet, I will bet again 1.000 USD next time…
Is this system going to work for me in the long run? Will I have at least 50% more ( 1.500.000 USD) after 5 years?
Is it possible to calculate it using probability and statistics theory? It should be… Let's say I only need to a confidence interval of 95%… I am OK with that…
Best Answer
We can put a probability on Barcelona winning at least once before you run out of money to fund the martingale strategy:
$ 10^6 = 2^i$ therefore $i = log_2(10^6) = 19.93157 \approx 19$
So you have 19 bets before you run out of cash.
Your profit percentage (given fractional odds) is $P=(OS_i - S_{i-1})/S_i$, where $i$ is the bet number you win on, $S_i$ is your stake, and $O$ are the odds (assumed constant). Since stake is given by $S_i = 2^{i-1}$ your percentage profit halves for every additional bet you have to take.
As time goes on, you're taking less and less rewarding bets. If bets are assumed independent from each other, your probability of winning nothing until the 19th bet are $p(1-p)^{18}/19$, which for $p=0.5$ is close to zero. However, your profit percentage would be roughly $1/10^6$ which is also tiny. It would correspond roughly to taking a bet at fractional odds of 1.000002 and betting a million dollars. Given that betting odds are not actually fair, if you took this bet enough times you'd lose money overall. House wins again :) Further, it's far worse than your return if you just put a million dollars in the bank.
Hope that helps.