Hopefully somebody will be able to help me resolve this debate. Also, please bear with me if i explain things in a circuitous way.
If you graph the odds of getting any given number in 36 rolls of 2 6 sided dice, you will get a V, as follows
5 ..................* * *
4...............* * * * *
3...........* * * * * * *
2.......* * * * * * * * *
1 * * * * * * * * * * *
.. 2 3 4 5 6 7 8 9 10 11 12
So, according to the graph, if you roll 2 6 sided dice 36 times, you will see 2 and 12 once each, 5 and 9 4 times each, and you will see 7 6 times, and so on.
Another way to say the exact same thing is as follows: in order to see a 2, both dice must land on 1. But in order to see a 3, Die1 must land on a 1 and die2 must land on a 2 <b>OR</b> die 2 must land on 1 and die1 must land on 2 so there are 2 chances for a 3. In order to see a 4, the possibilities are 1,3 or 3,1 or 2,2 so a total of 4 showing up by adding the product of rolling 2 dice has 3 possibilities... and so on.
Now, if you are rolling 3 dice and you wanted to graph that, it gets more complicated. Rolling a total of 3 has a chance of 1/218 getting a 4 is 3/218 getting a 5 is 6/218 getting a total of 6 is 10/218 and while i have not figured it out, i suspect the odds of getting a total of 7 is 15/218. But what happens if you graph the odds for rolling 3 dice like we did with the 2 dice?
6.......* * *
5.......* * *
4.......* * *
3....* * * *
2....* * * *
1 * * * * *
.. 3 4 5 6 7
It no longer graphs as a straight line, but now it graphs as a curve.
Now, here comes the debate... My buddy says that if you were to roll 4 dice it would graph as a bell curve, I agree. He then goes on to say that whatever shape a graph of the probability of rolling 4 dice are, that will be an identical graph as if you were to roll 2 dice twice as long, I disagree.
I think that if a person wants to make that argument, they are ignoring a crucial point. Specifically they are ignoring the fact that If you have a hypothesis that talks about adding the product of 2 separate dice rolls, among a very large number of dice rolls, you have to include in your equation the way you are deciding which two numbers to add.
For example, say you roll 2 dice 72 times and you want to take your 72 results and pair them up to create 36 new products. You could pair up the first and the second roll, the 3rd and the 4th roll... and so on, Or you could pair up the 1 and the 37th roll and the 2 and the 38th roll and so on... or you could make up your own system. However, as soon as you create a system, you have added new data to the equation that will produce an exponential result. So, what I end up with is this...
I am looking for a proof that shows that the equation for calculating the odds of rolling 2 dice is linear and an equation for calculating the odds for rolling 3+ dice is exponential and that they are fundamentally different equations and that rolling 2 dice for a longer time never approaches the same shape of a graph that rolling 4 dice would.
Any help would be appreciated.
Perhaps if something has a 1/36 chance of happening, the odds of it happening twice in a row are 1/1296 and if something has a 1/6 chance of happening, the odds of it happening twice in a row are 1/36. But i need more solid proof here, not just intuition.