This work studies the Pliable Index CODing problem (PICOD), a variation of the classical index coding problem where a client is satisfied if it can successfully decode at least one massage not present in its side information set. PICOD models `content-type coding' applications, such as web searches. PICOD significantly differs from classical index coding (where each client desires a specific message not present in its side information set) in the following sense. Past work showed that for PICOD O(log n) broadcast transmissions suffice to satisfy all the n clients, as opposed to the Ω(n) transmissions required by index coding; the key to this exponential improvement in number of transmissions is a probabilistic argument to show that a single transmission can satisfy at least a constant fraction of the clients. This paper elaborates further on this key result as follows. (i) A non-probabilistic analysis provides a lower bound, no smaller than 1/e, on the largest fraction of PICOD clients that can be satisfied by a single transmission in the case where all side information sets have the same cardinality; the new bound is tighter than known ones for any number of messages and clients, and sheds light into how the cardinality of the side information sets affects the number of clients satisfied by a single transmission. (ii) The same argument applied to the case where a message is in the side information set of any given client with probability p ∈ (0, 1) independent of all other messages and clients, and when there are sufficiently many messages, provides an more refined characterization of the performance than known results.