Penney's game, also called Penney-Ante, is a game of coin tossing in which two players select different three-symbol strings
of heads H and tails T. The coin is tossed until one chosen string
first appears as three consecutive outcomes, and its player wins. The outcomes are
independent random variables, with
H and T each having probability 
(Gardner 1974, p. 122).
If the first player chooses 
, the second can choose 
, where the bar interchanges H and T. This response
wins with probability at least 
and at most 
. Thus there is no unbeatable three-symbol string,
and the relation "more likely to appear first"
is not transitive (Gardner 1974, pp. 122-123).
More generally, consider two distinct strings 
and 
of the same positive integer length 
, using the same fair model of coin
tossing. For two such strings 
and 
, let 
be 1 when the last 
symbols of 
equal the first 
symbols of 
, and 0 otherwise, and define
 |
Writing 
for the event that 
appears before 
, the probability is
 |
The formula is due to Conway (Gardner 1974, pp. 123-124; Miller 2026).
