Is there a law that says if you do enough trials, rare things happen? I'm trying to make a video about loaded dice, and at one point in the video we roll about 200 dice, take all the sixes, roll those again, and take all the sixes and roll those a third time. We had one die that came up 6 three times in a row, which is obviously not unusual because there should be a 1/216 chance of that happening and we had about 200 dice. So how do I explain that it's not unusual? It doesn't quite seem like the Law of Large Numbers. I want to say something like "If you do enough tests, even unlikely things are bound to happen" but my partner said people might take issue with the "bound to" terminology.
Is there a standard way to state this concept?
 A: I think that your statement "If you do enough tests, even unlikely things are bound to happen", would be better expressed as "If you do enough tests, even unlikely things are likely to happen".  "bound to happen" is a bit too definite for a probability issue and I think the association of unlikely with likely in this context makes the point you are trying to put over.
A: I think what you need is a zero-one law. The most famous of these is the Kolmogorov Zero-One Law, which states that any event in the event space we're interested in will either eventually occur with probability 1 or never occur with probability 1. That is to say, there is no grey area of events that may happen.
A: Law of truly large numbers:

... with a sample size large enough, any outrageous thing is likely to happen.

A: You could explain that even as an event specified a priori, the probability that it occurs is not low. Indeed, it's not so hard to calculate the probability of 3 or more rolls of sixes in a row for at least one die out of 200.
[Incidentally, there's a nice approximate calculation you can use - if you have $n$ trials there there's a probability of $1/n$  of a 'success' (for $n$ not too small), the chance of at least one 'success' is about $1-1/e$. More generally, for $kn$ trials, the probability is about $1-e^{-k}$. In your case you're looking at $m = kn$ trials for a probability of $1/n$ where $n=216$ and $m=200$, so $k = 200/216$, giving a probability of around 60% that you'll see 3 sixes in a row at least once out of the 200 sets of 3 rolls.
I don't know that this specific calculation has a particular name, but the general area of rare events with many trials is related to the Poisson distribution. Indeed the Poisson distribution itself is sometimes called 'the law of rare events', and even occasionally 'the law of small numbers' (with 'law' in these cases meaning 'probability distribution').]
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However, if you didn't specify that particular event before the rolling and only say afterward 'Hey, wow, what are the chances of that?', then your probability calculation is wrong, because it ignores all the other events about which you'd say 'Hey, wow, what are the chances of that?'. 
You've only specified the event after you observe it, for which 1/216 doesn't apply, even with only one die. 
Imagine I have a wheelbarrow full of small, but distinguishable dice (maybe they have little serial numbers) - say I have ten thousand of them. I tip the wheelbarrow full of dice out:
die #    result
00001      4
00002      1
00003      5
 .         .
 .         .
 .         .
09999      6
10000      6

... and I go "Hey! Wow, what are the chances I'd get '4' on die #1 and '1' on die #2 and ... and '6' on die #999 and '6' on die #10000?"
That probability is $\frac{1}{6}^{10000}$ or about $3.07\times 10^{-7782}$. That's an astonishingly rare event! Something amazing must be going on. Let me try again. I shovel them all back in, and tip the wheelbarrow out again. Again I say "hey, wow, what are the chances??" and again it turns out I have an event of such astonishing rarity it should only happen once in the lifetime of a universe or something. What's up?
Simply, I am doing nothing but trying to calculate the probability of an event specified after the fact as if it had been specified a priori. If you do that, you get crazy answers.
A: Glivenko–Cantelli theorem (Wikipedia Link)
This theorem says, loosely speaking, that as the number of samples grows, the empirical distribution tends toward ("converges") the true distribution.
In that sense, if there truly is a nonzero probability of an event happening, enough observations should lead to you seeing it happen, since your empirical CDF has to tend toward the true CDF that gives such an event positive, even if small, probability.
