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?
 A: I agree with @Keaton - there are further examples in, for instance, best of three competitions, where commentators (specific examples are Artosis and Tasteless in StarCraft :P) will often tout 75% win rates after having won the first match. However, this is an example of an implication with completely probablistic interpretations - if you flip three coins, the probablity of getting two heads given the first coin coming up heads is exactly 75% (HHH, HHT, HTH, HTT).
(I know this should be a comment, but I don't know how to do that :( sorry)
A: This is not really an answer per se, but is too long for a comment.  This is a story I heard in my undergraduate Probability class which is possibly apocryphal, but I really have no idea.  It involves what one can legitimately infer from a sample data.  The story goes like this:
A study used monkeys and gave them pairs of colored M&Ms, using only Red, Blue, and Green.  They would give the monkeys one pair, and allow them to choose one.  Say the monkey was first given Red and Blue, and chose Red (or in other terms rejected the Blue M&M).  Then they gave the monkey a Blue and Green M&M.  They discovered that 2/3 of the time, the monkey would reject the M&M they rejected in the first pair (in our example case, the Blue M&M).  The conclusion they came to was that if the monkey rejected one color, it was more likely to reject the same color the second time.  So they made a behavioral conclusion from the test.  
But if you just assume that the monkey has a preferential order of colors of M&Ms, then it turns out that running through the combinations gives exactly a 2/3 probability of this happening.  That is, assume that the situation above happens, the monkey is given R,B, and rejects B.  Then it is given B, G.  What is the probability it will choose G over B?  All the possible orders are
RGB, RBG, GRB, GBR, BRG, BGR
And given that the monkey prefers R over B, there is a 2/3 chance it prefers G over B (the choices BGR, BRG work and the only other valid choice is GBR).
Maybe not what you are looking for, but it is a fun example of how not to infer too much from simple data.
A: I will tell you the problem have chosen in the end: The Monty Hall problem
A: 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
