How do you prove probability? I have this thought that has bugged me for a couple of days and could not find any answers on the internet. I thought this would be the best place to consult.
How does one go about proving probability? My family members are frequent casino goers and I was thinking of a way to show them how they won't ever win. Until I came across this problem myself.
Lets take a normal dice for example. There is a 1/6 chance of getting a random number from 1-6. But how can anyone be sure of this? Meaning there is 1 in 6 chances that the dice you threw will land on the number "1", but it could also be 10 in 60, 100 in 600, 1000 in 6000 and so on. So you may actually get any number other than "1" for the first 50 throws, and then getting a "1" on the next 10 throws to give you a 1 in 6 probability.
But OK, lets say you get lucky and you do hit a "1" in 6 throws. But why do you determine that is the probability of the throws? If you throw it for another 6 times, you may hit "1" two more times, giving you 3 in 12 and if you stop right there, the probability then is 1/4 or 1 in 4. If you don't get any "1" then your probability becomes 1 in 12. 
So who really determines/proves that the probability of a random number of a dice thrown is 1/6? Who decides how many times you have to throw it?
Just FYI all, I am no mathematical genius in probability and this is probably explained by some theory that I have never come across in my life. Would appreciate if you can point me in that direction if so. Thank you all.
 A: This is about modeling the reality. 
Mostly, we believe a die shows the $1$ approximately on in six times because this is how it always was observed. Also, good dies are sufficiently symmetric. To preferably land on one side this side has to be somehow special. If there is no special side there can be no reasoning for the die to land on it more than average.
Talking about probability, this is how we model random events, not how they are! We need to transfer them into the language of mathematics to calcaulate with them at all. So we throw the die, say, 10.000 times and together with above symmetry assumptions we see that it lands on each side sufficiently close to $1/6$ of the time. This confirms our hypthesis. 
Probability is decided in the long run. So when you throw a die twelve times and you got three $1$'s, then this might set you on the wrong track. You empirical result indeed shows a probability of $1/4$, but it has a very high chance to just be wrong. Your sample size is to small to safely do any statistics with it.
It all comes down to hypothesis testing. You will never know for sure that the chance os $1/6$ for each side. But you will get more certainty (in a strict mathematics sense) by throwing more and more dies. The chance to get three $1$'s during twelve throws is comparably large for a fair six-sided die. However, the cance to throw 3000 $1$'s during 12.000 throws is vanishingly small. So if you observe this behavior you are more likely to reject the assumption that you die is fair than to believe that this happended by pure chance.
A: You're correct, that, when dealing with statistics, only realizations of the possibilities are observable, but not the probabilities themselves.
One thing you can do is to look at the distribution of the dice throws, i. e. answer the question "After I did 60 throws, how many of them ended up 1, 2, 3, 4, 4, 5." This doesn't tell you a lot about the probablity, but you might get an idea of how the results ended up. For a 10-sided die, this might look like this: A Histogram from Wikipedia.
Also you can ask the question: Given these observed dice throws, how likely is it that it's a fair die (i. e. all sides do have the same probablity)? These kind of questions are answered by so-called Hypothesis Tests.
TL; DR: Histograms and Hypothesis Tests allow you to connect the empirical observations with the theoretic model assumptions.
