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A run is a sequence of more than one consecutive identical outcomes, also known as a clump.Let be the probability that a run of or more consecutive heads appears in independent tosses of a coin (i.e., Bernoulli trials). This is equivalent to repeated picking from an urn containing two distinguishable objects with replacement after each pick. Let the probability of obtaining a head be . Then there is a beautiful formula for given in terms of the coefficients of the generating function(1)(Feller 1968, p. 300). Then(2)The following table gives the triangle of numbers for , 2, ... and , 2, ..., (OEIS A050227).SloaneA000225A008466A050231A050233123456781100000002310000003731000004158310000531198310006634320831007127944720831082552011074820831The special case gives the sequence(3)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..

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