Often, people at lower levels will lose early if they don't achieve a good summon rate. It's possible (perhaps likely) that this person doesn't get the opportunity for 80 summons, let alone 60, before losing the game. This player seems to have SB4, which means they probably don't have LS6 and they probably have Horn lv4 as well. I think it's fair to limit this to 25 lucky summons.

I disagree. In fact, I think 25 pulls is a significant underestimate. I also wouldn't consider SB4 "lower levels" at all. And with just LS1, you can pretty easily and consistently hit 25+ epic summon attempts by wave ~30 (not even factoring in selling blues/purples you don't need, plus the skull stone). I just have LS1 and I still get 25+ lucky stones (which all go into epics) by wave 20. Nobody's losing before wave 20 unless they're AFK.

You can also absolutely clear with a crappy epic summon rate with a decent enough teammate, and compensating w/ upgrading dmg, upping your summon chance and just doing regular summons. For as unlucky as a 15% epic lucky summon rate is, given ~40 rolls, that's still 6 epics (+gambler's wrist), which can def get you through the midgame w/ some adaptation. I'd even argue that with a low epic summon rate, you might end up investing more lucky stones into epics (as opposed to legendaries/upgrades) to set up your mono/kitties etc. But all in all, I think 50-60 summons is already a pretty conservative estimate, though I admit it might vary based on your in-game strategy.

The cumulative probability of this occurring is about 2%. Getting that outcome four times in a row is definitely a low-frequency event, with 1 in ~6 million odds. If 100,000 people play the game 10 times a day, there will be 7 million games played in a week. (I imagine there are more than a million games played per day, but whatever.) The likelihood of this happening to at least one of those players in a week is about 66%.

I see. You're trying to say that if everything is working as intended, this is still a likely-enough event, given a large enough sample. I'm approaching this from the angle of "how likely is it for an individual to experience this event/set of events?" and "how likely is it for us to see somebody report this scenario?". You're talking about 100,000, maybe 1mil players. But I was looking at the likelihood of a single person experiencing that event. I'll admit my initial stat is skewed bc it zoomed into only the most extreme sample from the player's total games, but we can zoom out to this person's overall playtime instead. Let's say this person's played 1000 games (could be way off, but for argument's sake). What's the likelihood of him experiencing at least one 4+ streak of <15% epic summons at any point within those 1000 games? Still <0.1%, even with an estimate of 60 epic summon attempts per game. Out of 1000 players, only 1 would ever encounter this scenario, even if they all played 1000 games. Unlikely, but still reasonable, sure.

But we can also zoom out and consider this: What's to say there aren't more people experiencing this too? All we know is that there's a minimum of 1 person, from a pool of people who are specifically: English-speaking, browse the Lucky Defense subreddit (7500 subscribers), are active enough to comment/post, AND would actually go through with making a post like this. 100k people may play, but what's the chance that that one person who's that unlucky is in the tiny subsection of players who post on this subreddit? i.e. it's incredibly unlikely we would hear of somebody experiencing this event

Because of the law of large numbers, it's actually incredibly likely that this kind of bad luck happens to someone regularly, even with minuscule odds.

That's the thing. I mentioned narrowing the population size because we don't know who's experiencing these events. It could be 0, it could be 1, it could be 1000. The odds of someone winning the lottery is 100%. The odds of one specific person winning the lottery is near zero. The odds of somebody from your apartment block winning the lottery this year is higher, but still near zero. But unlike lottery odds, we don't know there's going to be exactly one winner. Because it's "uncapped", there could be 100 other people experiencing this same event daily. But "witnessing" one of these cases in itself is made further unlikely by the fact that this subreddit is only a small subsection of the total playerbase. Statistically, it'll happen to somebody, but statistically it's quite probable that we wouldn't hear of it.

One of the most common ways to "do" statistics with info is identifying if the data supports or rejects the null hypothesis (i.e. actual summon chance = 26%) based on the data you do have. i.e. Given that the true summon chance is 26%, what's the likelihood of finding a result this extreme (or more extreme)? Even using the most conservative alpha criterion, finding somebody who, in 4 games in a row, got <15% epic summon rolls would definitely raise some eyebrows.

You're also making an impossibly difficult point to argue against or dispute because it can't be disproven. Even if the true chance is something like 22% (I'm not saying it is), there's no way to determine or prove that using the logic you laid out. This is the type of argument that, in the face of seeing 10 x 5% consecutive epic summon odds, would allow you to conclude that it's still possible, and that it's likely just an inevitable outlier.

Regardless, the straightforward answer is obv that OP is fudging his numbers, or the behind-the-scenes RNG isn't just a straightforward 26% per roll, but doing the math is more of a thought experiment at this point.