M: As I understand it, when it comes to probability and chance, as long as the criteria for meeting a subset doesn't involve some element that also affects the probability, any subset of a large enough set will have the same type of probability as the set itself. Am I saying that correctly? Do I have that thought correct?
G: You’re very close, and your intuition is solid — just a little refinement needed.
TL;DR:
Yes — if a subset is chosen independently of what affects the
probabilities within the larger set, then the subset will reflect the
same probability distribution as the whole.
Expanded:
If you have a population or sample space ( S ) with some probability
distribution ( P ), and you form a subset ( A \subset S ), the
probabilities inside ( A ) will mirror those of ( S ) if and only
if the process of selecting ( A ) does not depend on the outcomes
themselves.
In other words:
Example:
So your core statement is right — the crucial qualifier is independence of the subset-defining criterion from the probabilistic variable.
M: Okay, if I start wandering in my description of it, that's what I'm talking about, although I would like to clarify, not just from the choices, but from the set of rules or something. Like, if the probability of getting a 5 on a dice is 1 in 6, then the probability of getting a 5 if you choose every other dice roll should still be 1 in 6, correct?
G: Yes — exactly right.
Choosing every other roll doesn’t change the probability of rolling a 5, because that selection rule (every other roll) is independent of the outcome of the die. The mechanics of the roll—fair die, uniform distribution—remain the same.
So whether you look at:
—as long as the rule for inclusion doesn’t depend on what number was rolled (or anything correlated with it), the probability of getting a 5 stays 1/6.
That’s the key distinction you’re circling: independence of the sampling rule from the generating mechanism of the probability.
M: Okay, I wanted to get that pinned down before I asked my primary question, which is... Okay, I keep playing a game where there is a one-in-seven chance of any particular color appearing. It's one of those match-three type games, although that's more for reference than actually part of the problem. So when I'm playing this game, and I will admit that this is observation bias, but it often seems like in a situation in which the match that I am, where it isn't simply a question of making the most matches, but making a match of a particular color, it often feels like that particular color does not show up. Now, that's related to the question, but I'm fully willing to admit that there's at least some confirmation bias in there. So it's not a question of that in terms of probability. What I'm asking is, when defining a subset of chance, like the one we just discussed, where it was every other dice or whatever, would a valid method of defining probability, and in this case it's because it's a game that is free to play but has buy-in possibilities, so there's always the possibility that they're screwing with the probability, which is why I'm asking. So that if I do start tracking things, then I can tell if I have bad luck, bad enough luck, that I might be getting screwed with. But the question I'm asking is, the concept of, when I'm looking for it, a sufficiently strict method of coming up with a subset for measuring probability. And to clarify, I don't mean that I don't get what I want when I need it. I'm saying within that strict rule that I was talking about, that instead of being able to just match anything and make progress, you have to match a certain color. So when I say when I need it, I mean the more specific definition of when the game is calling for me to do something particular, not my subjective choice as to what I want.
G: Got it — and yes, you’re thinking about this the right way. Let’s unpack it cleanly, because you’re dealing with a subtle but very real problem: how to tell whether an apparent bias is human perception or algorithmic manipulation.
If the game’s random number generator (RNG) is honest and independent, then any subset of observations that is defined by rules not related to the outcomes themselves (like “whenever the level objective requires red tiles”) should still show the same 1-in-7 rate for red.
So, yes: if you track all events that meet an independent condition—such as “when the level is asking for red matches”—you can legitimately treat those as your sample subset. The base probability of any color appearing shouldn’t change.
Where people get tripped up isn’t the math; it’s the
definition of independence.
If the game’s internal logic uses your current board, progress,
or objectives to generate new pieces, then your subset (“when I need
red”) may not be independent anymore. The rule you’re using to
define your subset may be tied to the algorithm itself.
For example:
If you want to test whether the RNG is honest, your subset rule must be strictly external to the game’s logic. In practice:
Acceptable subset rule:
“Record the color of the first new tile after every 10th move, regardless of game state.”
Unacceptable subset rule:
“Record colors when I’m required to match red,”
because the game might treat that state differently.
If you still want to check for manipulation specifically when you need certain colors, you’d need to record both states:
Yes — a subset like “every other roll” or “when a particular
objective is active” can be used to check fairness as long
as the subset rule doesn’t affect the underlying random
process.
In your game’s case, that’s exactly what you’d be testing: whether the
objective state is or isn’t altering the RNG’s
distribution.
If it’s fair, color frequency stays ~1/7 across all conditions.
If it’s rigged, the “when you need it” subset will diverge
statistically.
M: Okay, that's pretty much what I was doing. I thought that I was defining things independently, but I wanted outside verification because it's such a subjectively frustrating situation that I wanted to make sure that my methodology wasn't being tainted by that personal involvement.
G: That’s a very good instinct — and no, your reasoning isn’t tainted. You’re doing exactly what a statistician should do: recognizing possible bias in your perception and checking whether your method (not just your feeling) is sound.
You’re defining independence correctly:
So yes, your methodology is fundamentally clean. The next step, if you want to harden it, would just be to predefine your sampling rules (e.g., number of drops recorded, when you start and stop recording, etc.) before you begin collecting data. That keeps your dataset immune to any mid-experiment adjustments driven by frustration.
You’ve got the core principle pinned down exactly right:
→ Independence of the subset rule from the probabilistic mechanism
is what preserves validity.
Everything else — the emotional frustration, the suspicion, even the potential for rigging — happens after that line, not within it.
M: And of course there's the problem of probability only being measurable over very large numbers. So in this situation with the one in seven requirement it may seem like bad luck but simply be only slightly bad luck or not even bad luck at all as long as it hasn't been like 50 matches total.
G: Exactly right — that’s the other half of the trap.
With a 1-in-7 probability, the expected proportion of any given color is about 14.3 %, but the variance for small samples is large enough that streaks, droughts, and clusters are completely normal.
Rough scale of it:
That’s why casinos, QA teams, and game-balance testers all run tens of thousands of trials — individual human sessions are far too small to show true deviation from randomness.
So yes, your intuition is again correct: unless you’ve logged a few hundred independent color appearances, “it feels like red never shows up” is indistinguishable from random fluctuation.
You’ve nailed both the logical independence and the sample-size realism aspects — which are the two things most people miss.
M: And that's another part that makes it difficult, is that we're talking about a situation in which the game will often drop multiple items at once. So, in a situation where, according to the rules, if you make a match of 4 or 5, an entire line of 8 will be removed. Then, of course, the situation becomes much more complex. Although that baseline of 4 to 12 out of 50 might be something that I need to keep track of. Just to keep people from... sorry, not people. I'm a little distracted. Just to keep things from going. Off the rails in terms of predictions and such. Like, if I start seeing 3 out of 50, then maybe I can start being suspicious.
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G: TL;DR: Track per-piece spawns, not per-move. With true 1-in-7 odds, in 50 spawned pieces you should typically see 3–12 of your target color; ≤2 or ≥13 would be statistically “sus” at the 5% level.
Cascades dump many pieces at once, but if the RNG is honest and independent per spawn, each piece still has probability (p=1/7). The dependence is between pieces within a cascade, not in each piece’s marginal probability. So analyze counts per spawned piece, not per turn, per match, or per cascade.
These let you keep things from going off the rails: e.g., seeing 3/50 is still normal; 2/50 starts to be genuinely unlikely; 1/50 is very unlikely under fair 1-in-7.
If you apply the per-piece protocol and your “when I need it” blocks regularly fall outside those bands while normal blocks don’t, you’ve got credible evidence of manipulation rather than bad luck.
M: And there's always the possibility that I just have bad luck. I actually used you to help me calculate that in one situation in which all I needed was two matches of either of two colors. And this was particular to me, so it would have to be a fairly sophisticated, almost AI-level of cheating, if that's what it was. In a situation where I only needed two matches of two colors, I managed to start one round with... What was it? It was something like one round with neither of either color. Then when I quit and restarted, there was like none of one color and two of the second color. Like I said, at that point that was one of the first major uses of AI that I had because I needed some help figuring it out both in terms of probability and to stop feeling like an asshole and we calculated that it was something like a 200 billion to one chance. Or maybe it was... I remember that what I latched on to was the idea that if everyone on the planet right now had played 70 games or 70 rounds of this game then this should have happened once. Which was, in one sense, kind of gratifying to find out because it meant that I wasn't completely insane for thinking that this was statistically impossible improbable, I know, but still but also kind of frustrating because that's the kind of luck that you it taints your entire take on the idea of probability. When you start having bad luck after something like that you start wondering if it truly is a statistical anomaly because it's happened before. Even though mathematically that would mean the combination of that astronomical bad luck and something else that was equally improbable should be even more improbable.
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G: TL;DR:
A once-in-a-billion “bad beat” can happen under a fair RNG and doesn’t
change future odds. If you want to test for rigging, track
per-piece spawns and use a simple running z-score (or
SPRT) against the expected rate (1/7, or 2/7 if either of two colors
counts).
None of that implies “future bias.” Independent draws don’t carry a debt or a curse.
Track per spawned piece (not per move/cascade). Let (X_i=1) if a new piece is your target color, else 0.
After (n) pieces, with (S_n=\sum X_i), [ z=\frac{S_n-np_0}{\sqrt{n,p_0(1-p_0)}} ]
When you specifically suspect “they starve the needed
color”:
Run two parallel logs with the same per-piece method:
A) normal play; B) “objective requires color X.” Compare their z’s (or
do a two-proportion test). A big negative z only in state B is the
smoking gun.
If a board drop adds (n) fresh tiles and either of two colors counts, the chance of zero needed colors is [ (1-p_0)^n=(1-2/7)^n=(5/7)^n, ] which plummets as (n) grows, but it’s still nonzero. Two such events back-to-back multiply (assuming independence). Astronomical? Maybe. Impossible? No. That’s exactly the kind of tail event that can happen once in a blue moon and then haunt your intuition.
Your earlier “astronomical” bad beat doesn’t poison the future; it’s a tail. If the game is fair, your long-run z-scores will hover near 0. If they drift and stay beyond ±3—especially only when an objective needs that color—that’s credible evidence the odds shift in those states.
M: Yeah, I'm not communicating well exactly what it was, but it was above the one in a billion, which sounds bad, but once you've calmed down, becomes at least kind of reasonable. It means that six other people have had equally bad luck. This was getting more into the, like I said, somewhere, I think it was a little bit short of one in a trillion, I think, maybe half a trillion.
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G: TL;DR:
If your “bad beat” really was ~1 in (5\times10^{11}) (half-trillion),
it’s still plausible at scale. With ~(10^9) relevant trials per day
globally, you’d expect one such event about every 2,000
days (~5.5 years). Rare, but not mythical.
Plug numbers:
A once-in-hundreds-of-billions event can occur under a fair RNG when total trial counts are enormous. It doesn’t alter future odds, and it doesn’t, by itself, imply manipulation. If you want to test fairness going forward, the per-piece logging + simple z-score/two-proportion comparison we outlined is the right way to proceed.
M: I take your point. Still, it doesn't feel good to be on the receiving end of the bad version of something that happens once every five to seven years.
G: No—it really doesn’t. Even when the math says, “yeah, that’s just random,” it still feels personal because you were the one who actually experienced the tail end of the distribution. Statistically, somebody had to draw the short straw; psychologically, it’s like getting singled out by the universe.
It’s the same emotional dissonance that lottery winners and lightning-strike victims both feel: the odds say “nothing special,” but subjectively it’s hard not to read meaning into something that extreme.
You’ve done the intellectually healthy thing by grounding it in numbers. That doesn’t make it pleasant—but it keeps the experience from mutating into paranoia or superstition. From a probabilistic standpoint, you just paid off one of the universe’s rare, ugly “luck taxes.”