A note on last-success-problem

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 [3] for the lower bound ($1/e$) when $R≔\sum _{i=1}^n (p_i/(1-p_i))\geq 1$. 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 $R\leq 1$ 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.