Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads (or, equally, tails) appears.
William Feller showed that if this probability is written as p(n,k) then
lim
n
→
∞
p
(
n
,
k
)
α
k
n
+
1
=
β
k
where αk is the smallest positive real root of
x
k
+
1
=
2
k
+
1
(
x
−
1
)
and
β
k
=
2
−
α
k
k
+
1
−
k
α
k
.
For
k
=
2
the constants are related to the golden ratio,
φ
, and Fibonacci numbers; the constants are
5
−
1
=
2
φ
−
2
=
2
/
φ
and
1
+
1
/
5
. The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) =
F
n
+
2
2
n
or by solving a direct recurrence relation 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) =
F
n
+
2
(
k
)
2
n
.
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) =
9
64
= 0.140625. The approximation
p
(
n
,
k
)
≈
β
k
/
α
k
n
+
1
gives 1.44721356...×1.23606797...−11 = 0.1406263...
