More on n-point, win-by-k games

1996 ◽  
Vol 33 (2) ◽  
pp. 382-387 ◽  
Author(s):  
John Haigh
Keyword(s):  
Bernoulli Trials ◽  
Probabilistic Argument ◽  
Domain Of Applicability ◽  
Markov Dependence

When Siegrist (1989) derived an expression for the probability that player A wins a game that consists of a sequence of Bernoulli trials, the winner being the first player to win n trials and have a lead of at least k, he noted the desirability of giving a direct probabilistic argument. Here we present such an argument, and extend the domain of applicability of the results beyond Bernoulli trials, including cases (such as the tie-break in lawn tennis) where the probability of winning each trial cannot reasonably be taken as constant, and to where there is Markov dependence between successive trials.

More on n-point, win-by-k games

1996 ◽  
Vol 33 (02) ◽  
pp. 382-387 ◽  
Author(s):  
John Haigh
Keyword(s):  
Bernoulli Trials ◽  
Probabilistic Argument ◽  
Domain Of Applicability ◽  
Markov Dependence

When Siegrist (1989) derived an expression for the probability that player A wins a game that consists of a sequence of Bernoulli trials, the winner being the first player to win n trials and have a lead of at least k, he noted the desirability of giving a direct probabilistic argument. Here we present such an argument, and extend the domain of applicability of the results beyond Bernoulli trials, including cases (such as the tie-break in lawn tennis) where the probability of winning each trial cannot reasonably be taken as constant, and to where there is Markov dependence between successive trials.

A limiting distribution for Bernoulli trials with second-order Markov dependence

1983 ◽  
Vol 20 (2) ◽  
pp. 419-422 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Second Order ◽  
Limiting Distribution ◽  
Bernoulli Trials ◽  
Markov Dependence

Let ℰ be an event governed by a homogeneous second-order two-state Markov chain. Assume the steady state has been achieved. Consider Nm the number of times that ℰ occurs in m successive trials. The limiting distribution of P(Nm = k) is obtained under the condition that mP(ℰ)=λ.

A limiting distribution for Bernoulli trials with second-order Markov dependence

1983 ◽  
Vol 20 (02) ◽  
pp. 419-422 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Second Order ◽  
Limiting Distribution ◽  
Bernoulli Trials ◽  
Markov Dependence

Let ℰ be an event governed by a homogeneous second-order two-state Markov chain. Assume the steady state has been achieved. Consider Nm the number of times that ℰ occurs in m successive trials. The limiting distribution of P(Nm = k) is obtained under the condition that mP(ℰ)=λ.

On Ramsey Minimal Graphs

10.37236/1184 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Tomasz Łuczak
Keyword(s):  
Infinite Number ◽  
Probabilistic Argument ◽  
Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest $F$ other than a matching, and every graph $G$ containing a cycle, there exists an infinite number of graphs $J$ such that $J\to (F,G)$ but if we delete from $J$ any edge $e$ the graph $J-e$ obtained in this way does not have this property.

The Bernoulli Trials 2004

2006 ◽  
Vol 79 (3) ◽  
pp. 199-205
Author(s):  
Christopher G. Small ◽  
Ian Vanderburgh
Keyword(s):  
Bernoulli Trials

Binary decompositions of probability densities and random-bit simulation

2020 ◽  
Vol 26 (2) ◽  
pp. 163-169
Author(s):  
Vladimir Nekrutkin
Keyword(s):  
New York ◽  
Distribution Function ◽  
Random Number ◽  
Discrete Distribution ◽  
Random Number Generation ◽  
Academic Press ◽  
Bernoulli Trials ◽  
Algorithms And Complexity ◽  
Binary Decomposition ◽  
Probability Densities

AbstractThis paper is devoted to random-bit simulation of probability densities, supported on {[0,1]}. The term “random-bit” means that the source of randomness for simulation is a sequence of symmetrical Bernoulli trials. In contrast to the pioneer paper [D. E. Knuth and A. C. Yao, The complexity of nonuniform random number generation, Algorithms and Complexity, Academic Press, New York 1976, 357–428], the proposed method demands the knowledge of the probability density under simulation, and not the values of the corresponding distribution function. The method is based on the so-called binary decomposition of the density and comes down to simulation of a special discrete distribution to get several principal bits of output, while further bits of output are produced by “flipping a coin”. The complexity of the method is studied and several examples are presented.

n-point, win-by-k games

1989 ◽  
Vol 26 (4) ◽  
pp. 807-814 ◽  
Author(s):  
Kyle Siegrist
Keyword(s):  
Exit Time ◽  
Bernoulli Trials ◽  
Expected Number ◽  
First Exit Time ◽  
World Series ◽  
Time Space ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Positive Integers ◽  
Made In

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

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