Simple probability problems which hide important concepts Together with a group of students we need to compose a course on probability theory having the form of a debate. In order to do that we need to decide on a probability concept simple enough so that it could be explained in 10-15 minutes to an audience with basic math knowledge. Still, the concept to be explained must be hidden in some tricky probability problems where intuition does not work.
Until now we have two leads: 


*

*the probability of the union is not necessarily the sum of the probabilities

*Bayes' law (for a rare disease the probability of testing positive when you are not sick is very large)
The second one is clearly not intuitive, but it cannot be explained easily to a general audience. 
Do you know any other probability issues which are simple enough to explain, but create big difficulties in problems when not applied correctly?

There are three prisoners in Cook Maximum security prison. Jack, Will and Mitchel. The prison guard knows the one who is to be executed. Jack has finished writing a letter to his mother and asks the guard whether he should give it to Will or Mitchel. The prison guard is in a dilemma thinking that telling Jack the name of a free person would help him find out his chances of being executed. 
Why are both the prisoner and Jack wrong in their thinking?
You have 4 conditions
Jack     Will      Mitchel   probability go free     probability of jack being executed
X        jail       jail            Will                        1/6
jail      X         Jail            Mitchel                     1/6
jail      Jail       X              Will                        1/3
X        jail       jail            Mitchel                     1/3
