I want to introduce the concept of expected value (for a discrete random variable, finite case) in a high school class. My first idea was to start with a simple game of chance and let the students play it. Suppose there are 25 students. If everybody plays the game once I have 25 outcomes which I could take to calculate the mean value of profits for this game. Then I want to discuss it and go over to calculate the probabilities of the game and then introduce the concept of expected value.
My first idea for a game was this one:
Throw a fair dice twice (= two random integers $i$ and $j$ between 1 and 6). If $i = 4$ and $j = 4$ you win 4\$, if exactly one of them is 4 you win 2\$ if $i \neq 4$ and $j \neq 4$ you lose 1\$.
The expected value for this game turns out to be -0,03\$.
However if you simulate this by playing the game 25 times you never end up with a mean value close to this, even if I let each student play it two times, such that I get 50 values it is most times far away from the mean value. I simulated it with a little
python script and noticed that you have to play it at least 1000 times to come close to some extent to the expected value, so this example converges too slow for my purpose.
So is there another game (of similar complexity) which converges very much faster and which is such that I can play it with the students as described above before discussing it?