Feller's coin-tossing constants are a set of numerical constants which describe

asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...

that in ''n'' independent tosses of a

fair coin In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In th ...

, no run of ''k'' consecutive heads (or, equally, tails) appears.

William Feller William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian- American mathematician specializing in probability theory. Early life and education Feller was born in Zagreb to Ida Oemichen-Perc, a Croa ...

showed that if this probability is written as ''p''(''n'',''k'') then :

\lim_ p(n,k) \alpha_k^=\beta_k

where α_''k'' is the smallest positive real root of :

x^=2^(x-1)

and :

\beta_k=.

Values of the constants

For

k=2

the constants are related to the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

\varphi

, and

Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

; the constants are

\sqrt-1=2\varphi-2=2/\varphi

and

1+1/\sqrt

. The exact probability ''p''(n,2) can be calculated either by using

, ''p''(n,2) =

\tfrac

or by solving a direct

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...

leading to the same result. For higher values of

k

, the constants are related to generalizations of Fibonacci numbers such as the tribonacci and tetranacci numbers. The corresponding exact probabilities can be calculated as ''p''(n,k) =

\tfrac

. Coin Tossing at WolframMathWorld
/ref>

Example

If we toss a fair coin ten times then the exact probability that no pair of heads come up in succession (i.e. ''n'' = 10 and ''k'' = 2) is ''p''(10,2) =

\tfrac

= 0.140625. The approximation

p(n,k) \approx \beta_k / \alpha_k^

gives 1.44721356...×1.23606797...⁻¹¹ = 0.1406263...

References

{{Reflist

External links

Steve Finch's constants at Mathsoft
Mathematical constants Gambling mathematics Probability theorems