# Two thirds law

The two-thirds law , also the law of the third or the law of small numbers , is a sentence from stochastics that describes a special case of binomial distribution in the case of small success probabilities of randomly caused events. This term is mostly used in connection with the game of roulette and describes the fact that about two thirds of the 37 numbers are hit in 37 games.

The name Law of Small Numbers goes back to the Russian-German mathematician Ladislaus von Bortkewitsch (1898), who found this law when investigating the number of deaths from hoofbeats in the individual cavalry units of the Prussian army.

## The law of two thirds in roulette

If you look at several rotations in roulette, i. H. Series of 37 individual games each ( French coups ), it is found that only about two thirds of the numbers are hit in the course of a rotation, about half of them several times, while the remaining third is not hit - hence those of roulette players used terms two-thirds law or, more rarely, law of the third .

In the course of a rotation in roulette, the mean

• 36.3% of the numbers, i.e. H. 13.4 numbers not hit
• 37.3% of the numbers, i.e. H. 13.8 numbers hit exactly once
• 18.6% of the numbers, i.e. H. 6.9 numbers hit exactly twice
• 6.0% of the numbers, i.e. H. 2.2 numbers hit exactly three times
• 1.7% of the numbers, i.e. H. 0.6 numbers hit four or more times.

Note : These values ​​were calculated using the binomial distribution , which models the problem mathematically exactly. The fact that the sum of the listed values ​​does not result in exactly 100% or 37 figures is due to the rounding. Often the problem is analyzed using the Poisson distribution , which gives a relatively good approximation.

According to the law of large numbers , each of the 37 numbers occurs with the same relative frequency as a long-term mean. H. if the number of coups is sufficiently large , each individual number has the same share, namely 1/37 = 2.7%. If you look at several rotations and a number determined in advance, this is hit once on average in each rotation.

This leads many players to the wrong conclusion that in a series of 37 spins every single number occurs once. But this is not the case; rather, it is extremely unlikely that each number will be hit exactly once; the probability of this is only 1.3 · 10 −15 .

Despite the equal probability of all numbers, there is no equal distribution for the 37 roulette series, understood as a Bernoulli chain with 37 repetitions, but the above pattern given by the binomial distribution.

Even with the help of the two-thirds law, no winning strategy can be found (see Marche ).

## The general case

The law of small numbers is a simple application of the Poisson distribution for and applies not only to a series of 37 roulette games, but to any series of independent games, each of which can have equally probable outcomes (see Poisson approximation ). So z. B. when objects are raffled among recipients and the individual draws are independent of each other . ${\ displaystyle \ lambda = 1}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$

The larger the number, the more precisely the law of small numbers applies . For , the proportion of recipients who receive exactly objects strives against the value ${\ displaystyle n}$${\ displaystyle n \ to \ infty}$${\ displaystyle k}$

${\ displaystyle P (X = k) = {\ frac {1} {k!}} \; {\ rm {e}} ^ {- 1}}$

with Euler's number . ${\ displaystyle e}$

The proportion of recipients who do not get anything therefore tends towards . The same applies to the proportion of those who are considered exactly once. ${\ displaystyle e ^ {- 1} \ approx 36 {,} 7879 \, \%}$

The figures for given in the previous section only deviate by 0.5% from the limit values ​​calculated using the Poisson distribution. ${\ displaystyle n = 37}$

## Example grains of rice

Grains of rice randomly scattered on the ground

The picture on the right shows grains of rice lying randomly on the ground. Image section and grid size are chosen so that on average a grain of rice falls on a square, i.e. H. it applies . ${\ displaystyle \ lambda = 1}$

Counting the frequencies confirms the expected values (approximated using the Poisson distribution) despite the small sample size : ${\ displaystyle n = 64}$

• 23 squares contain no grain of rice (red). Expected value (rounded to 2 decimals): 23.54
• 25 squares contain exactly one grain of rice (yellow). Expected value: 23.54.
• 12 squares contain exactly two grains of rice (green). Expected value: 11.77.
• 2 squares contain exactly three grains of rice (blue). Expected value: 3.92.
• 2 squares contain four or more grains of rice (purple and gray) (1 × 4 or 1 × 5). Expected value: 1.22.

(The sum of the expected values, rounded to one decimal, is 64.0.)

## Individual evidence

1. Ladislaus von Bortkewitsch: The law of small numbers. Leipzig 1898 ( online )
2. a b Jörg Bewersdorff : Luck, Logic and Bluff: Mathematics in Play - Methods, Results and Limits , Springer Spectrum , 6th edition 2012, ISBN 978-3-8348-1923-9 , doi : 10.1007 / 978-3-8348 -2319-9 , p. 61.
3. ^ Matthias Strunz: Maintenance: Basics - Strategies - Workshops , Springer Vieweg, 2012, ISBN 978-3-642-27389-6 , doi : 10.1007 / 978-3-642-27390-2 , p. 221.