A full-information best choice problem is considered. A sequence of N iid random variables with a known continuous distribution function is observed. The number of observations N is a positive random variable 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 kth 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.