TITLE: Example for a simple game of chance where the average value converges fast to the expected value
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: I did something similar to this just today.
Roll three 6-sided dice.  If at least one 6 is rolled, you win \$1.  If not, you lose \$1.  This is a nice one because you may be able to "convince" students that the game is fair: the probability of getting a 6 on one die is 1/6, there are 3 dice, and $3 \times 1/6 = 1/2$.  But in fact the probability of winning is only about 42%.
I had 13 groups and asked each of them to play 10 times, and aggregated the 130 total results.  (I believe there were 52 wins, about 40%.)  With this much data the standard deviation is about 4.3%, so it's very unlikely you will be anywhere close to 1/2, and you can with high confidence reject the null hypothesis that the game is fair.  This kind of parallelizing is a nice way  to gather a lot of data fast.
This is similar to the carnival game chuck-a-luck and its relative sic bo.  There are other payoff schemes possible, which generally still result in a quite unfavorable game.