Law of the Third is an observed tendency for a third of the numbers on the roulette wheel to not appear during a set number of spins. There are 37 numbers on a single-zero roulette table, but after 37 spins we are likely to see that 24 numbers have landed while 13 – about a third – did not. This is a mathematical fact but not an official mathematical law; it is mostly only mentioned in the game of roulette. To be a successful and winning player, you need to know all the available options and types of bets that you can make at the roulette table. Read all the Roulette Rules.

Let’s see if this information can be used to create a roulette betting system or is it another fallacy that will cause losses.

**Law of the Third Mathematical Formula.**

In any random draw of 37 different numbers for 37 times, it is statistically expected that 36.285% of the numbers will not be drawn at all, while others will be drawn once or more than once.

The formula for this is 36/37 to the 37th power. The result is 0.36285, and that’s the chance of one number not landing at all, so it’s therefore also a chance for all numbers to not land at all.

Meaning, of 37 numbers on a single-zero roulette wheel, we’re likely to see 13.63 numbers that didn’t land at all and 23.37 that landed once or more than once. If we only look at the 36 numbers, excluding zero, then there should be 13.06 numbers that didn’t land and 22.94 that did, or roughly 13:23.

More specifically, in 37 spins there’s an expected distribution of 13 numbers that do not hit, 11 numbers that hit once, and 13 numbers that hit twice. These results are prone to fluctuation but overall, over a large sample of trillions of spins, within a sequence of 37 spins, there will be between 14 and 34 numbers that did hit. The Law holds true; it’s just a matter of lower and upper limit within one sample.

For comparison, it would take an infinite number of spins to find one sequence of 37 spins where 37 numbers all landed exactly once. That does not happen in real life through the Law of Large Numbers says this is the tendency.

**Does the Law of the Third apply to the physical third of the wheel?**

If we were to divide the roulette wheel into three sections – or would use a three-sided bet such as 1st third / 2nd third / 3rd third, the Law of the Third would still apply, but with different parameters. Instead of 36/37 to the 37th power, the formula would be 2/3 to the 3rd power, and the result would be the chance of one-third of the wheel not landing within three spins. The chance would be the same as in the previous example but with lower volatility.

If we were to use six spins as a sample, the chance of one-third of the wheel not landing would be reduced to 8.8%. Note: in this example that talks about one-third of the wheel, the zero were not accounted for, and we used three groups of 12 numbers for simplicity.

**What to do with this information?**

If we were to use the Law of the Third to try to make money on roulette, we would likely want to place a bet on those numbers that didn’t yet land, as we may think they are “due” to land sooner or later.

It will definitely happen sooner or later, but the question is when. Law of the Third is true, but still, we have no way of knowing what will happen on the next spin as each spin is completely individual and it has no memory of previous spins. We also can’t know if we’re betting on a number that is among those that will not show up. If we were to use the Law of the Third to bet on numbers that didn’t land, we would show a misunderstanding of the Law of Large Numbers and would be guilty of Gambler’s Fallacy.

One thing we can do with this knowledge is understanding that it’s natural for a third of the numbers to not appear at all within 37 spins, and therefore we’d be wiser than the average gambler who, if questioned, would state that within 37 spins all 37 numbers are statistically expected to land exactly once.

And then, at the very least, we wouldn’t be led to believe that we have just observed a statistical anomaly we can exploit. And could perhaps build a decent roulette system using this knowledge.

**Roulette Systems Based on Law of the Third.**

Since the Law of the Third is true and is expected to be observed at any roulette table, there are systems that can be built that use this statistical information in one way or another, and they would likely fare better than those based on the belief that all numbers must land at least once in 37 spins.

In order to use the Law of the Third for predicting future spins we’d also have to believe that the roulette wheel has memory and knows which numbers are “due”, so it yet again constitutes Gambler’s Fallacy, but at least here we’re working with formulas that describe what actually happens in real life, as opposed to the Law of Large Numbers that describes the general theory that has no practical application.

There’s no foolproof system in roulette, but ones that use or at least acknowledge the Law of the Third look better than those that choose to ignore it even if we can’t know whether our number(s) will be among those that come up or among those that do not.