Probability via Geometry, applications and examples People, I am going to teach a lesson including something about Probability via Geometry and, if you have, I would like to know some references or materials (or even some good ideas) that can help me. The target public to this lesson are young guys witch ages between 14 and 17 years old, so I would like to talk about basics (and as intuitive as possible) results witch the proof can be understood for the public.
In fact, the subject of this class must to be "strategies to teach probability concepts via geometry".
Can somebody help me?
 A: My favorite example is as follows: 
QUESTION
Buses leave the terminal station exactly in every $2$ hours the whole day. If a man arrives to the terminal station at a random instant then the average waiting time is $1$ hour -- quite clear, isn't it?
At the $10^{th}$ bus stop, $100$ miles away from the terminal station, the average arrival (and leaving) time is still $2$ hours. How come, then, that for the man arriving randomly to the $10^{th}$ bus stop the average waiting time is $1.5$ hours?
ANSWER
At the terminal bus station the the leaving instants are distributed evenly as shown in the upper figure below. So, the probability that the man gets to the terminal station in the $i^{th}$ interval is independent from $i$.

At the $10^{th}$ station the buses arrive and leave less evenly because of unpredictable events taking place on the road from the terminal station. There are shorter and longer intervals. The probability that the man catches a longer interval is higher than catching a shorter interval. As a result, the average waiting time is longer.
This example can be elaborated depending on the knowledge of the audience.
