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.

Amazingly, *spinning* a penny instead of tossing it results in heads only about
30% of the time (Paulos 1995).

Diaconis *et al. *(2007) proposed that flipping a coin introduces a small degree of wobble which causes the coin to spend more time in the air (before landing) with
the initial top side facing up. As a result, a "same-side bias" is introduced
so that a coin is slightly more likely to land with the side initially facing upward
before the toss on top after landing. Diaconis *et al. *(2007) estimated a "same
side" probability for a fair coin toss of 51% (as opposed to exactly 50%) based
on a "modest" number of empirical observations. This prediction has been
confirmed experimentally by Bartoš *et al. *(2013), who collected a total
of 350,757 coin flips and found that the probability of the initial side facing upwards
upon landing was 50.8% with a 95% confidence interval
of 50.6%-50.9%. The data also showed no trace of a heads-tails bias, with 175,420
heads obtained from 350,757 tosses, giving a heads probability of 50.0% with a 95%
confidence interval of 49.8%-50.2% (Bartoš
*et al. *2013).

There are some rather counterintuitive properties of coin tossing. For example, for a fair coin toss, 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).

The probability of a coin of finite thickness landing *on an 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.

coin | diameter (mm) | thickness (mm) | |

penny | 19.05 | 1.52 | 1/5900 |

nickel | 21.21 | 1.95 | 1/3800 |

dime | 17.91 | 1.35 | 1/7000 |

quarter | 24.26 | 1.75 | 1/8100 |

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).