I recently came across two smallish examples of where randomness was the solution to two perplexing problems. That is, rolling the dice seems to help you out of a situation where a planned method was not giving you what you wanted.
The first issue is the problem of how to board passengers on a plane. Finding the best way to board people is actually a well-studied problem, both theoretically and in practice, and you can see some of the work here. At the top of the same page there is a nice simulation program which shows you how different boarding strategies play out, and random boarding (just calling out people to board at random) is better than the usual front-to-back boarding that most of us are familiar with.
The reason is that random boarding gives a better utilization of the space in the plane whereas front-to-back boarding piles people into one part of the plane, eventually causing jams in the aisles. The full set of strategies examined are
On another topic, a Freakonomics blog post describes how researchers in South Africa are using a randomness trick to get truthful answers from farmers who are suspected of illegally killing leopards and hyenas. The method is called randomized response surveying, where when the farmers are asked potentially incriminating questions they first flip a coin, and based on the result give a yes or no answer to either the incriminating question if it was heads, or a harmless question (do you think the Springboks will win the RWC?) if it was tails. The farmers actually used a die, taking specific actions on which value from 1 to 6 was thrown, but the principle is the same as I have described it.
The trick here is that the person asking the question cannot tell which question the farmer is answering, but the farmer’s answer can be recorded. Statistical methods can then be used to determine the distribution of answers for the two questions, and actually make inferences about the proportion of positive answers to the incriminating question. This method was devised in the 60’s, and by the early 80’s it was being taught at my undergraduate university as part of a first year course.