On reaching head-to-tail ratios for balanced and unbalanced coins
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چکیده
Given a coin with head-to-tail probability ratio 1 to with integer ¿ 1. Let T denote the time it takes to reach the head-to-tail ratio r= q=( q + m). We prove that the limit probability P(T ¡∞) of ever reaching this ratio r is ( + 1)=(m + + 1), as q → ∞ and q and m are co-primes. Asymptotic results for the conditional expected time E(T |T ¡∞) and standard deviation (T |T ¡∞) are also derived. A surprising aspect of the results is that these quantities are not continuous functions of the ratio r. c © 2002 Published by Elsevier Science B.V. MSC: 60C05; 05A15; 33C20
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