# Coin Tossing — from Wolfram MathWorld

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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

distribution.

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 *H*s and

*T*s 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

where

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.

RED : CANON VIETNAM COMPANY LIMITED CR0004852

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

*k*-step number.

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).

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