Tænkeboks gives a good and thorough description of how you can get from a description of a simple game of dice to an influence diagram that can help find a winning strategy. The network is the result of a 7th semester project at Aalborg University, Denmark, and the documentation is written by one of the participating students.
Asia is a small Bayesian network that computes the probability of a patient having tuberculosis, lung cancer or bronchitis respectively based on different factors - for example whether or not the patient has been to Asia recently.
Poker is a Bayesian network that models a simplified version of the game of poker. The network can help predict who has the best hand - your or your opponent.
The Mildew examples are four different influence diagrams modeling the situation where a farmer has to decide on a treatment with fungicides for a wheatfield. The influence diagrams model different scenarios depending on the information available.
Implementations of these examples come with the installation of any version of Hugin. If you have lost them or have an older installation of Hugin, you have the opportunity to download them.
If you have made some examples of your own, you are welcome to put them at forum.hugin.com. Here you can also find discussions, links and other things related to the field of Bayesian networks.