On the waiting time in a generalized roulette game
This paper considers the following problem: a roulette has numbers 0,1,2,...,m-1 as possible outcomes; the game continues until the first time that numbers 1,2,...,k have all appeared before 0 appears; if a zero appears too soon, the game starts again from scratch. A generalization of this problem considers a multinomial experiment with outcomes \(E_ 0,E_ 1,...,E_{N+1}\); the experiment is repeated until \(E_ 1,...,E_ N\) have all occurred before \(E_ 0\) occurs. Let S be the total time elapsed until the game ends; expressions for the expectation and the variance, as well as for the moment generating function of S are obtained. Also, some asymptotic results are derived for S as N tends to infinity: in the particular case where \(E_ 0,...,E_ N\) have all the same probability \(1/(N+1)\), and \(E_{N+1}\) has probability 0, for example, \(S/(N+1)^ 2\) converges in distribution to an exponential random variable with parameter 1. The authors then go back to the original roulette problem, and to the problem of rolling two six-sided dice repeatedly, and determining the number of rolls necessary to obtain 4,5,6,8,9 and 10 before 7 is obtained, and apply their general formulas to these two particular cases.
scientific article; zbMATH DE number 3876314