The Mirage of the Martingale
Posted by silentarchimedes on May 17, 2008
Like all those people looking to find the edge in Roulette at the casino, I fell into what I call “the Mirage of the Martingale“. The Martingale’s premise is in putting money only on 50/50 bets, like the flip of a coin. In Roulette, the closest to this is the RED/BLACK bet, ODD/EVEN bet or 1-18/19-36 bet. It is not a full 50/50 bet because the two green slots (0 and 00) reduces each bet to a 47.3684% winning probability instead of the 50% winning probability in 50/50 bets. (Remember, American Roulette has 38 slots, 18 for RED, 18 for BLACK, and 2 for GREEN. This creates a 2.6316% for landing on any specific slot. The house edge is the two GREEN slots, or 2*2.6316 = 5.2632%)
The basic strategy of the Martingale is when you lose a 50/50 bet, you double your bet on the next spin. That way, if you win, you cover your last round’s loss and also gain your initial bet’s amount.
For example, let’s say you bet $5 on BLACK in round 1. The spin turns up RED. You are down $5. In round 2, you now bet $10 on BLACK. If the spin turns up black, you cover the $5 you lost in the first round, and you also make $5. If you lose, you are now down $15. In round 3, you now bet twice the last round’s bet, or $20. And so on and so forth… Assuming you lose 5 BLACK bets in a row until winning on the sixth, your betting sequence is 5, 10, 20, 40, 80, 160, … and your profit sequence is 0, -5, -15, -35, -75, -155, 5.
In short analysis, this sounds like a fool proof idea because you assume that if you keep betting on black, at some point it will land on black, and you will cover your losses.
Upon further analysis, there are some obvious problems with the Martingale. The problems are both in the mathematics, and in the practicality of using such a strategy in a casino.
1. First, the mathematics. At first glance, the probability of a losing streak of 5 seems very small. For every spin, there is a 1/2 probability of winning or losing. Five in a row has a 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32 = 3.125% likelihood. If someone asked you if you would bet $5, and the chance of you losing is 3% and the chance of you winning is 97%, you would take that bet in an instant. However, this 3% is nice when you are talking short term (although you might be unlucky). For example, if you stay at the table for 5 rolls, your chance of hitting that losing streak is a small 3%. However, the longer you stay at the table, the more you are exposing yourself to the Law of Large Numbers and the Central Limit Theorem (see images below, courtesy of Wikipedia).
|As the number of trials increases, the output plot starts to resemble a bell curve. Even the unlikely outcomes eventually occur, based on their respective probabilities. In gambling, unwanted outcomes will eventually show up if enough bets are placed.|
Eventually, that 3% will show its ugly face. This is also true for losing streaks greater than 5, such as 6=1.56%, 7=0.78%, 8=0.39%, etc. On a positive note, as the length of a losing streak increases, the probability decreases and you can stay longer at the table without fear of the two above theorems/laws. For example, the chance of a losing streak of 8 is only 0.39% for any 8 straight rolls. However, a losing streak of 8 means you would have to wager $640 on the 8th round, and this is all an attempt to get your original $5 back!! A person with a limitless purse would not care, because they know eventually they will break this losing streak and cover however big their loss is. Unfortunately, the casinos know this! Which leads to our second problem…
2. Every casino has a minimum bet and maximum bet on the roulette table. A typical “cheap” table usually has a minimum of $5, and a maximum of $100 or $200, or for short (5,100) and (5,200) table. When I started going to casinos at 18, I always wondered why the maximum bets are so low. Now I know. The maximum limit of a (5,200) table would limit a person using the Martingale strategy to a maximum losing streak of 6. That means if you bet $160 on that sixth spin and lose, you lose $315. You cannot bet $320 to potentially cover your losses because that is over the max limit of the table. You must carry that $315 loss with you!! All for an original $5 bet! That risk proposition doesn’t seem so good now.
These two problems might not fully convince some people to not use the Martingale strategy. The first thought is I will make enough money before I hit a losing streak of 6. Now we talk about how to bet on winning streaks and why they are not practical to overcome the odds of hitting a losing streak. The most common method is the Anti-Martingale Strategy:
The Anti-Martingale Strategy - The betting strategy is the same as the Martingale except you double your bet when you win. However, you must have winning streak level where you bank your profit, and then begin betting at $5 again. Let’s say you make your level at 4. So when you win 4 in a row, you bank 5, 10, 20, 40 = $75. Not bad huh? The problem is a winning streak of 4 only occurs 1/2 * 1/2 * 1/2 * 1/2 = 1/16 = 6.25% Remember, a (5,200) table only allows a max losing streak of 6. If it happens, you lose $315. As mentioned, these losing streaks occur 1.56% of the time. That means win streaks of 4 occur exactly 4 times more than losing streaks of 6. Both of these are the only two instances in which non-even money is exchanged. All other streaks are break even for you. So, for every time you lose $315, you only make 4 * $75 = $300! You are still short $15. And all these computations above don’t include the house edges on the greens. Which would mean a slightly larger loss on your end! No matter how you change the length of the winning streak when you bank your money, you are on the short end. For example, say you bank every time you win two in a row = $15. This occurs 25% of the time. This occurs 16 times more than your losing streak of 6. That means for every time you lose $315, you make only (16 * 15) = $240!
Our LS6-WS4 Martingale/Anti-Martingale Strategy on a (5,200) American Roulette Table looks like this:
I tried modifying the Martingale and Anti-Martingale and even ran simulations in Matlab. The numbers never ever add up. In the end, it still comes down to luck.
Anybody disagree or have ideas about at least slowing down your losing in a mathematically dependable way? I apologize if the math is somewhat off, as I did this on the spot, but I think no matter what, it’s impossible to use the Martingale effectively in a casino.
**Update 05/21/08 ** – Yesterday’s NBA lottery draft is a perfect example that anything can happen, even with the probability against you. Probability is simply that, the likelihood of something happening and not happening. If something has a nonzero probability (even if it’s 1.7%), it can still happen at anytime. That’s what happened last night. The Chicago Bulls, had a 1.7% chance of landing the #1 pick in the draft. Eight teams were ahead of them (in terms of probability). Miami had the highest probability to land the top pick, with 25%. So guess what happens? That 1.7% happens, and the Bulls get the top pick!! (As a Knicks fan, I was rooting for the Knicks 7.6% chance, but instead of picking even 5th, they ended up 6th)