The dice may have no memory, but probability wins in the end.
That's probably the best way to describe the law of large numbers, which states that as you repeat the same experiment, coin flip, dice roll, etc., the results will get closer and closer to the expected value. Basically, the more times you repeat something, the closer you will get to the truth. You may get 6 heads in a row when you flip a coin, but if you flip 1,000 times, you'll probably be very close to 50/50.
In fact, just for fun, I went to Random.org's Coin Flipper, which let me flip 100 coins at once. (Yes, this is what I do for fun.) After the first 100 flips, I had 46 heads, 54 tails. After 500 flips, I had 259 heads (51.8%) and 241 tails (48.2%). If I kept flipping, I would have eventually gotten to 50/50.
We use this probability theory in statistics and research all the time. Because as our sample size gets closer to the population size, the closer our results will be to the true value (the value we would get if we could measure every single case in our population). Relatedly, as our sample size increases, the closer the sample distribution will be to the actual population distribution. So if a variable is normally distributed in the population, then as sample size increases, our sample will also take on a normal distribution.
While that magic number (the sample size needed) to get a good representation of the population will vary depending on a number of factors, the convention is not as big as you might think - it's 30.
That's right. Recognizing that all kinds of crazy things can happen with your sample (because probability), 30 cases is about all you need to get a good representation of the underlying population. That information, however, shouldn't stop you from doing the unbelievably important step of a power analysis.
I'll be talking more about population and sample distributions, like those pictured, soon. Sneak preview - one of them has to do with beer.