I recommend the example in the book Doing Bayesian Data Analysis, in this book to motivate MCMC : A politicean want to visit all of US states and each with frequency proportional to the total population of the state. He takes decisions to visit a new state comparing the population of the current and the proposed state, He don't know in advance the total population of every state, but applying MCMC he archieves his goal. Here you may think that computing the total population of the US is a simple detail, but in other problem the computation involves a very difficult multivariable integral with thousand of variables, that in practice is impossible to calculate.