Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T19:44:58.704Z Has data issue: false hasContentIssue false
  • English
  • Français

Coin tossing, revisited

Published online by Cambridge University Press:  14 July 2016

O. E. Percus  and
J. K. Percus
O. E. Percus*
Affiliation:
New York University
J. K. Percus*
Affiliation:
New York University
*
∗ Postal address: Courant Institute of Mathematical Sciences
∗ Postal address: Courant Institute of Mathematical Sciences
Get access Rights & Permissions [Opens in a new window]

Abstract

An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current success ratio. We find the probability P1 that this estimate always lies above some pre-selected rational fraction p′, and its complement P2, the probability that it will reach p′ or below at least once. In the region p′ ≧ p, P1 = 0. In the region p′ < p, P1 ≠ 0 and is furthermore a discontinuous function of p′ at every rational p′.

Keywords

BERNOULLI TRIALSSURVIVAL PROBABILITY DISCONTINUITYLATTICE RANDOM WALK
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 70 - 80
Copyright
Copyright © Applied Probability Trust 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by D.O.E. DE-AC02-76ER03077.

References

[1] Bernoulli, J. (1713) Ars Conjectandi (Opus posth.). Basileae.Google Scholar
[2] De Moivre, A. (1711) De mensura sortis, seu, de probabilitate eventuum in lundis a casufortuito pendentibus, Phil. Trans. London 27, 213–264.Google Scholar
[3] Madras, N. TO be published.Google Scholar
[4] Markoff, A. A. (1912) Wahrscheinlichkeitsrechnung. Leipzig.Google Scholar
[5] Percus, O. E. and Percus, J. K. (1983) Phase transition in a four dimensional random walk with application to medical statistics. J. Statist. Phys. 30, 755–783.Google Scholar
[6] Percus, O. E. (1985) Phase transition in one-dimensional random walk with partially reflecting boundaries. Adv. Appl. Prob. 17, 594–606.Google Scholar
[7] Percus, O. E. and Percus, J. K. (1987) Piecewise homogeneous random walk with a moving boundary. SIAM J. Appl. Math. 47, 822–831.Google Scholar
[8] Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton, NJ. (2nd edn. Springer-Verlag, 1976).Google Scholar
[9] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
[10] Uspensky, J. V. (1937) Introduction to Mathematical Probability. McGraw-Hill, New York.Google Scholar
[11] Wald, A. (1947) Sequential Analysis. Wiley, New York.Google Scholar
[12] Wax, N., (ed.) (1954) Selected Papers on Noise and Stochastic Processes. Dover, New York.Google Scholar