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. A coin toss corresponds to a Bernoulli Distribution with . 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
*HTT*. Furthermore, it is six times as likely that *HTT* will be the first of *HTT*, *TTH*, and *TTT*
to occur (Honsberger 1979). More amazingly still, *spinning* a penny instead of tossing it
results in heads only about 30% of the time (Paulos 1995).

Let be the probability that no Run of three consecutive heads appears in independent tosses of a Coin. The following table gives the first few values of .

0 | 1 |

1 | 1 |

2 | 1 |

3 | |

4 | |

5 |

Feller (1968, pp. 278-279) proved that

(1) |

(2) |

and

(3) |

(4) |

(5) |

(6) |

(7) |

**References**

Feller, W. *An Introduction to Probability Theory and Its Application, Vol. 1, 3rd ed.* New York: Wiley, 1968.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/feller/feller.html

Ford, J. ``How Random is a Coin Toss?'' *Physics Today* **36**, 40-47, 1983.

Honsberger, R. ``Some Surprises in Probability.'' Ch. 5 in *Mathematical Plums*
(Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 100-103, 1979.

Keller, J. B. ``The Probability of Heads.'' *Amer. Math. Monthly* **93**, 191-197, 1986.

Paulos, J. A. *A Mathematician Reads the Newspaper.* New York: BasicBooks, p. 75, 1995.

Peterson, I. *Islands of Truth: A Mathematical Mystery Cruise.* New York: W. H. Freeman, pp. 238-239, 1990.

Spencer, J. ``Combinatorics by Coin Flipping.'' *Coll. Math. J.*, **17**, 407-412, 1986.

© 1996-9

1999-05-26