Martingale (betting System) - Alternative Mathematical Analysis

Alternative Mathematical Analysis

The previous analysis calculates expected value, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll.

As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll.

In reality, the odds of a streak of 6 losses in a row are much higher than the many people intuitively believe. Psychological studies have shown that since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely. This intuitive belief is sometimes referred to as the representativeness heuristic.

The odds of losing a single spin at roulette are q = 20/38 = 52.6316%. If you play a total of 6 spins, the odds of losing 6 times are q6 = 2.1256%, as stated above. However if you play more and more spins, the odds of losing 6 times in a row begin to increase rapidly.

• In 73 spins, there is a 50.3% chance that you will at some point have lost at least 6 spins in a row. (The chance of still being solvent after the first six spins is 0.978744, and the chance of becoming bankrupt at each subsequent spin is (1 − 0.526316)×0.021256 = 0.010069, where the first term is the chance that you won the (n − 6)th spin – if you had lost the (n − 6)th spin, you would have become bankrupt on the (n − 1)th spin. Thus over 73 spins the probability of remaining solvent is 0.978744 x (1-0.010069)^67 = 0.49683, and thus the chance of becoming bankrupt is 1 − 0.49683 = 50.3%.)
• Similarly, in 150 spins, there is a 77.2% chance that you will lose at least 6 spins in a row at some point.
• And in 250 spins, there is a 91.1% chance that you will lose at least 6 spins in a row at some point.

To double the initial bankroll of 6,300 with initial bets of 100 would require a minimum of 63 spins (in the unlikely event you win every time), and a maximum of 378 spins (in the even more unlikely event that you win every single round on the sixth spin). Each round will last an average of approximately 2 spins, so, 63 rounds can be expected to take about 126 spins on average. Computer simulations show that the required number will almost never exceed 150 spins. Thus many gamblers believe that they can play the martingale strategy with very little chance of failure long enough to double their bankroll. However, the odds of losing 6 in a row are 77.2% over 150 spins, as above.

We can replace the roulette game in the analysis with either the pass line at craps, where the odds of losing are lower q=(251:244, or 251/495)=50.7071%, or a coin toss game where the odds of losing are 50.0%. We should note that games like coin toss with no house edge are not played in a commercial casino and thus represent a limiting case.

• In 150 turns, there is a 70.7% chance that you will lose 6 times in a row on the pass line.
• In 150 turns, there is a 68.2% chance that you will lose 6 times in a row at coin tossing.

In larger casinos, the maximum table limit is higher, so you can double 7, 8, or 9 times without exceeding the limit. However, in order to end up with twice your initial bankroll, you must play even longer. The calculations produce the same results. The probabilities are overwhelming that you will reach the bust streak before you can even double your bankroll.

The conclusion is that players using martingale strategy pose no threat to a casino. The odds are high that the player will go bust before he is even able to double his money.

Contrary to popular belief, table limits are not designed to limit players from exploiting a martingale strategy. Instead, table limits exist to reduce the variance for the casino. For example, a casino which wins an average of $1000 a day on a given roulette table might not accept a $7000 bet on black at that table. While that bet would represent a positive expectation of over $368 (10/19 · 7000 − 18/38 · 7000 = 368.42) to the casino, it would also have a 47.37% chance of negating an entire week's profit.