Theory of Probability and Mathematical Statistics
A note on last-success-problem
J.M. Grau Ribas
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Abstract: We consider the Last-Success-Problem with $n$ independent Bernoulli random variables with parameters $p_i>0$. We improve the lower bound provided by F.T. Bruss for the probability of winning and provide an alternative proof to the one given in \cite {BR2} for the lower bound ($1/e$) when $R\coloneqq\sum_{i=1}^n (p_i/(1-p_i))\geq1$. We also consider a modification of the game which consists in not considering it a failure when all the random variables take the value of 0 and the game is repeated as many times as necessary until a ``$1$'' appears. { We prove that the probability of winning in this game when $R\leq1$ is lower-bounded by $0.5819\ldots=\frac{1}{e-1} $}. Finally, we consider the variant in which the player can choose between participating in the game in its standard version or predict that all the random variables will take the value 0.
Keywords: Last-Success-Problem, lower bounds, odds-theorem, optimal stopping, optimal threshold
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