An idealized coin consists of a circular disk of zero thickness which, when thrown in the air and allowed to fall, will rest with either side face up (“heads”
H or “tails” T) with equal probability. A coin is therefore a two-sided
die. Despite slight differences between the sides and nonzero thickness of actual coins, the distribution of their
tosses makes a good approximation to a Bernoulli
There are, however, some rather counterintuitive properties of coin tossing. For example, it is twice as likely that the triple TTH will be encountered before
THT than after it, and three times as likely that THH will precede
HHT. Furthermore, it is six times as likely that HTT will be the first
of HTT, TTH, and TTT to occur than either of the others (Honsberger
1979). There are also strings of Hs and
Ts that have the property that the expected wait to see string
is less than the expected wait to see , but the probability of seeing before seeing
is less than 1/2 (Gardner 1988, Berlekamp et
al. 2001). Examples include
1. THTH and HTHH, for which and
, but for which the probability that THTH
occurs before HTHH is 9/14 (Gardner 1988, p. 64),
2. , , but
for which the probability that TTHH occurs before HHH is 7/12, and
for which the probability that THHH occurs before HHH is 7/8 (Penney
1969; Gardner 1988, p. 66).
More amazingly still, spinning a penny instead of tossing it results in heads
only about 30% of the time (Paulos 1995).
The probability of a coin of finite thickness landing on edge has been computed by Hernández-Navarro and Piñero (2022) as
is the critical angle of the cylinder of radius and thickness comprising the coin. Remarkably, this expression
is independent of the coefficient of restitution. The chances of US coins landing
on an edge computed using this formula are summarized in the following table.
coindiameter (mm)thickness (mm)penny19.051.521/5900nickel21.211.951/3800dime17.911.351/7000quarter24.261.751/8100
The study of runs of two or more identical tosses is well-developed, but a detailed treatment is surprisingly complicated given the simple nature of the
underlying process. For example, the probability that no two consecutive tails will
occur in tosses is given by , where
is a Fibonacci
number. Similarly, the probability that no consecutive tails
will occur in tosses is given by ,
where is a Fibonacci
Toss a fair coin over and over, record the sequence of heads and tails, and consider the number of tosses needed such that all possible sequences of heads and tails of
length occur as subsequences of the tosses.
The minimum number of tosses is (Havil 2003,
p. 116), giving the first few terms as 2, 5, 10, 19, 36, 69, 134, … (OEIS
A052944). The minimal sequences for are HT and TH,
and for are HHTTH, HTTHH, THHTT, and TTHHT.
The numbers of distinct minimal toss sequences for , 2, … are
2, 4, 16, 256, … (OEIS A001146), which appear
to simply be .
It is conjectured that as becomes large, the average number of
tosses needed to get all subsequences of length is ,
where is the Euler-Mascheroni
constant (Havil 2003, p. 116).