A full-information best choice problem is considered.

A sequence  of $N$ {\em iid\,} random variables ({\em rv}'s) 

with a known continuous  distribution function ({\em df\,}) 

is observed.

The number of observations $N$ is a positive {\em rv\,} 

independent of observations.

The objective is to maximize the  probability  of  selecting 

the best (largest) observation when one choice can be made.

At each stage a solicitation of the present observation as well as

of any previous ones is allowed. If the $(k-t)$th observation with

the value $x$ is solicited at the $k$th stage, the probability of

successful solicitation may depend on $t$ and $x$.

General properties of optimal strategies are shown and

some natural cases are examined in detail.

Optimal strategies and their probability of success 

(selecting the best) are derived.