Probability that theoretical results match experimental results I am not sure if this can be determined, but I was wondering if there was any way to go deeper into probability to find the odds that your experimental results match your theoretical results.
For example, we know that if a fair die is rolled
 there is a 1 in 6 chance that it will come up 3 (arbitrary). So, if it is rolled 6 times, what are the odds that this will hold true and a three will come up exactly once?
Thanks
 A: We will assume that the theoretical model fits the reality well. This is the case for the dice example of the post. 
The probability of exactly one $3$ can be computed explicitly, since the number of $3$'s in $6$ tosses has Binomial Distribution (please see Wikipedia for details).
This probability is 
$$\binom{6}{1}\left(\frac{6}{1}\right)^1\left(\frac{5}{6}\right)^5,$$
where $\binom{6}{1}=6$. Computation shows that this is approximately $0.4$, not very big, but not terribly small.
If we toss $18000$ times, the probability of exactly $3000$ $3$'s is quite small. However, let $N$ be the number of $3$'s in $18000$ tosses. Then the ratio $\frac{N}{18000}$ will likely be quite close to $\frac{1}{6}$. One can state quite precisely how likely and how close. For example, with probability about $0.99$, the ratio $\frac{N}{18000}$ will be no more than $0.0072$ away from $\frac{1}{6}$. 
For a more detailed discussion of related matters, please see the Wikipedia article on The Law of Large Numbers. 
