For the basic Bayes Formula one common example to use is disease screening. Assume that you have a test for a disease that if used on someone who has the disease will show positive with 95% probability and if used with someone without the disease will show negative with 90% probability; further we know that 1 in 1,000 in the population have the disease. We randomly choose a person from the population (don't know ahead of time if they have the disease) and do the test which turns out positive: what is the probability that they have the disease? This example is often eye-opening to a lot of people. One way to demonstrate this (and quickly show the effect of changes) is using the
SensSpec.demo function in the TeachingDemos function for R (also see
tkexamp in the same package for a GUI interface to this in the examples).
If you want to expand to Bayesian statistics then one fun approach is to start by showing the students a simple success/fail game like throwing a dart at a target, tossing a wadded up piece of paper into a basket, etc., and choosing a student that will play the game. Ask the students how many times out of 4 they predict the student will succeed, and use their prediction as parameters for a Beta distribution as the prior distribution (plot this to show where they think the true probability could be). Now have the student do the game 10 times and count the successes, use this as the data for a binomial likelihood, and combine with the prior to get a posterior distribution for the student's proportion of successes. Show how you moved from a prior to a posterior using data and fairly simple calculations. If you have time you can let the student play the game more times and use the first posterior as a new prior, then get an updated posterior, and show how the distribution changes with additional information.