Maximizing a probability: A student workshop on an application of continuous distributions

For many students meeting, say, the gamma distribution for the first time, it may well turn out to be a rather fruitless encounter unless they are immediately able to see an application of this probability model to some real-life situation. With this in mind, we pose here an appealing problem that can be used as the basis for a workshop activity introducing, and subsequently encouraging the exploration of, many of the well-known continuous distributions in a meaningful way. We provide suggestions as to how the session might be run, discuss any pedagogical issues that arise and highlight particularly interesting features of the distributions.