I found an online blackjack that has no minimum or maximum bet limit. The dealer stands on soft 17. Also, in the event that both dealer and player gets blackjack, it is a push. There is no insurance policy and 21 on both sides are always a tie.
I have 10,000 credits to work with.
With the Martingale betting system, I figured that I am more likely to gain a profit than to lose credits. But I'm not quite sure how to calculate the math behind this and how to calculate the probability for winning a single blackjack hand.
I figured with 10,000 credits, I can start my bet at 1, then go to 2, then 4, etc (doubling my bet everytime I lose). This allows me to lose up to 12 hands before I am bankrupt. On the other hand, if I win any of these 12 hands, I gain profit and just reset my bet to 1 again. The way I see it, I have over a 50% chance to win at least one of these 12 hands, thus always profiting.
Am I favored to gain profit in this specific type of blackjack? How can I calculate the probability for this?
Best Answer
Whether the game is in your favor is independent of the betting system. No system of betting can rescue a losing game. You are correct that with Martingale you are more likely to win than lose a given series, but if the basic bet is losing the losses will be large and probable enough to give a negative expectation.
For your example, if your chance of winning an individual hand is $0.49.0.50,0.51$, your chance of losing $12$ in a row is about $0.000310,0.000244,0.000191$. If you lose $12$ in a row, you lose $4095$, so the expectation of a series is $-0.268,0,0.215$
You can't calculate the chance of winning a hand until you specify the strategy you will follow. It is still too complicated to compute by hand, so people resort to computer modeling. You use a random number generator to deal a lot of hands and count how many you win.