Coin tossing is an experiment whose usual outcomes are heads H and tails T. In the ideal fair model, each outcome has probability 
, and repeated tosses are Bernoulli
trials with outcomes that are independent
random variables. An idealized coin is a circular disk
of zero thickness, so it acts as a two-sided die. Actual
coins have nonzero thickness and slightly different sides,
and their tosses are often approximated by 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 is supported by the experiment of Bartoš et al. (2025), who collected 350,757 coin flips. They estimated the probability of landing with the initial side upward as 50.8%, with a 95% credible interval of 50.6%-50.9%. The degree of the same-side bias varied between people. When the initial side was randomly determined, there was no evidence of a heads-tails bias. The published data contain 175,421 heads, and the estimated heads probability is 50.0%, with a 95% credible interval of 49.8%-50.2% (Bartoš et al. 2025).
There are some rather counterintuitive properties of coin tossing. For example, in the ideal fair model, 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
expectation value 
of the wait to see string 
is greater than the expectation
value 
of the wait to see 
,
but the probability of seeing 
before seeing 
is greater than 1/2 (Gardner 1988; Berlekamp et al. 2004,
p. 885). 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, while the first occurrences of
THHH and HHH finish simultaneously with probability
7/8 (Penney 1969; Gardner 1988, p. 66).
For a cylindrical coin with vertical translational velocity and rotation about a horizontal diameter, Hernández-Navarro and Piñero (2022) give the probability of landing on an edge as
 |
where 
is the critical angle for a cylinder of radius 
and thickness 
. Their model assumes repeated partially inelastic bounces
with a constant coefficient of restitution, negligible air and ground friction, and
equiprobability of the relevant prior phase states. Within this model, the expression
is independent of the coefficient of restitution. The following table gives approximate
probabilities for US coins, with inverse probabilities rounded to the nearest hundred.
| 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. In the ideal fair model, for integers 
and 
, 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.
In this model, consider the number of tosses needed such that all possible sequences of heads and tails of length 
occur as contiguous blocks 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),
and equal 
.
This count follows by cutting a binary de
Bruijn sequence at any of its 
positions and repeating its first 
symbols. The number of cyclic sequences
up to rotation is 
(de Bruijn 1946, p. 758).
Let 
be the number of tosses needed for
all contiguous blocks of length 
to occur in this model. Its expectation
value satisfies
 |
where 
is the Euler-Mascheroni constant and
tends to zero as 
in little-O notation.
This follows by combining the exponential tail bound of Móri (1987, p. 112)
with the quantitative convergence theorem of Móri (1990, p. 916).
