A note on last-success-problems

TL;DR

This paper analyzes the Last-Success-Problem with Bernoulli variables, improves existing probability bounds, and explores game modifications where players can repeat or choose strategies, providing new theoretical insights.

Contribution

It offers improved lower bounds for winning probabilities and introduces new game variants with strategic options and their probabilistic analysis.

Findings

Lower bound for winning probability improved beyond previous results.

Derived an explicit lower bound for a modified game with repeated attempts.

Analyzed strategic choice between participating or predicting all zeros.

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 for the lower bound ($1/e$) when $R:=\sum_{i=1}^n (p_i/(1-p_i))\geq1$. 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 is lower-bounded by $e^{-1}(1-e^{-R})^{-1}$. 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.