u/immersedpastry

This was a question asked by u/yosh711, about which fails were Henry's favorite and least favorite!

Favorite Fail (First Image): Airship (Revenged). I'd imagine it must've been really satisfying for Henry to get to go out on his own terms for a change. He must've been pretty disappointed when he found out that it counted as a failure.

Least Favorite Fail (Second Image): Walkthrough (Toppat 4 Life). Henry has died so many times that he's in all honesty numb to it. Most deaths were physically painful, and pretty quick. But being stuck watching the same idiot for hours, potentially days, was mentally draining for him in a way that most of his other failures just weren't.

Hi, Reddit! I've been a lot more active on Tumblr lately, so if you like my art, feel free to check my stuff out at @immersedpastry!

Someone on Tumblr (original post) asked a question about how likely it would be for a person who had never played Henry Stickmin before to complete the game without failing or getting softlocked, so I thought I’d take a crack at it. The math behind it is actually a lot simpler than you'd think, even considering softlocks like that.

The Short Answer

With the model I constructed, the odds that they'd complete a perfect run are, at the absolute best, roughly 1 in 5 trillion against. On average, though, they'd be in the ballpark of 1 in 7-10 trillion against. Either way the hypothetical person is about 20,000 times more likely to hit the Powerball jackpot.

The Long Answer

When I did the calculations, I made two big assumptions about the fails, the first of which is that the probability of choosing any of the given options on a screen is uniform. So in the simplest case with only a single screen and n options (i.e. Breaking the Bank), the probability of choosing any given option is just 1/n. In particular, the odds of success for a screen with only one path forward is 1/n. Since screens are independent events, the probability of making it to a particular ending will be a product of the probabilities that the player chooses the option that advances them to that particular ending, irregardless of whether the others are fails or branching paths. Taking Toppat Civil Warfare as a relatively simple example, the player has a 1 in 4 chance of choosing Helicopter Hat, a 1 in 3 chance of choosing Dave, and a 1 in 3 chance of choosing Deuces, so the total odds of success are 1/4 * 1/3 * 1/3 = 1/36, or about 2.8%. The total odds of success for the game will be the sum the probabilities that the player reaches all of its endings, because the endings are mutually disjoint events.

The only tricky things are the quick time events. I didn't want to treat them as equally probable for a player to get. An ending with a lot of quick time events should be harder to avoid failing than others. For my second assumption, I decided to approximate the odds of timing out on a quick time event with exponential decay. For a quick time event lasting t seconds, the probability of timing out in the model is e^-kt, where k is a nonnegative coefficient that roughly correlates with the player's reaction time. At k = 0, the player is guaranteed to time out on every quick time event, with larger and larger values of k making success for even small values t very likely. The complement of this probability was multiplied by the odds of getting the correct option on a quick time event to get the odds of success for any given quick time screen.

I got the following results:

Breaking the Bank

16.7% (exactly 1 in 6) chance of success

Not much to say here since there’s just the one possible ending with the single option. But 83% of all the possible journeys have already ended.

Escaping the Prison

k = 0 case: 2.78% (exactly 1 in 36) chance of success
k = 1 case: 3.34% (about 1 in 30) chance of success
As k -> ∞, odds approach 5.38% (about 1 in 18)

Out of all successful games, Lawyered Up is by far the most common outcome. It's the only possible successful outcome when k = 0 since it has no QTEs, but even for larger values of k, no fewer than about 51.6% of them will be Lawyered Up. The percent of Badass Bust Out games tends toward 38.7% for large k, but only around 8.2% of games will be for k = 1. Sneaky Escapist becomes the least likely outcome for k > 1.034, accounting for around 8.5% of successful games for k = 1 and tending toward only 9.7% for larger values of k.

Stealing the Diamond

k = 0 case: 0.03% (exactly 1 in 3,456) chance of success
k = 1 case: 0.24% (around 1 in 419) chance of success
As k -> ∞, odds approach 0.35% (around 1 in 288)

Intruder on a Scooter is the most common outcome for large k. Although impossible at k = 0, 51.5% of all successful games end here at k = 1, going toward 2 out of every 3 for large k. The proportion of Just Plain Epic games is 36.4% at k = 1, but approaches 1 in 4 for large k. Unseen Burglar is least likely of all on average, accounting for 12% of games at k = 1 and tending towards 8.33%.

Infiltrating the Airship

k = 0 case: 0.0027% (exactly 1 in 36,864) chance of success
k = 1 case: 0.0167% (about 1 in 5,982) chance of success
As k -> ∞, odds approach 0.35% (around 1 in 288)

Government Supported Private Investigator makes up 20.6% of games when k = 1 and tends toward 18.8% as k increases. Rapidly Promoted Executive and Relentless Bounty Hunter are predictably equiprobable, both making up around 31.6% of games when k = 1 and going to 33.5% when k is large. Pure Blooded Thief is on average rarest but not by as significant a margin as in previous endings, accounting for 16.2% for k = 1, and 14.1% for large k.

Fleeing the Complex

Fleeing the Complex is where things get a little bit tricky. The softlock in IRO/TB isn't all that bad to work around, you just have to multiply the original product for both endings by the odds of not getting softlocked, which is the complement of the odds of getting PBT given that the previous game was successful. The really bad bit is Presumed Dead, because the final screen is only successful on a timeout of a quick time event lasting 14 seconds. The odds of getting that timeout are e^-14k, a really, really tiny number for large k. And for smaller k, it becomes less likely that the player will pass the previous quick time events. The end result is that Presumed Dead will only account for not much more than 0.0005% of all Fleeing the Complex endings, even at their highest concentration. This is a bit of a limitation of the model, so it's very likely that the true likelihood of getting Presumed Dead on a random playthrough as described is significantly higher. To be fair, though, up to that point no timeout has ever resulted in a fail, so there's definitely no way in practice that someone with no prior knowledge of the game would be able to see it coming without clicking on one of the other options first. So it's probably not that big a deal. Convict Allies accounts for 3 out of every 7 games that succeed for k = 0, and Ghost Inmate the remaining 4 out of 7. For k = 1, these figures dip to 3% and 4%, respectively, and remain like that for larger k. The vast majority of the remaining games are either International Rescue Operative at around 18.5%, and The Betrayed, at 74.3%.

Completing the Mission

Completing the Mission is more straightforward than I thought it would be, too. I worked out the probabilities of clearing each ending on their own, and then to get the final figures I multiplied the probabilities of clearing the first three games and the associated ItA and FtC endings to get the 1 in 5-10 trillion figure I arrived at before.

Interestingly, the overwhelming majority of games that succeeded ended in Revenged, accounting for a whopping 87.6% of all games. In fact, 37.6% of all games follow the same pathway, Lawyered Up -> Intruder on a Scooter -> Rapidly Promoted Executive -> The Betrayed -> Revenged, the most common out of any of them.

Following at distant seconds, thirds, and fourths are Toppat Civil Warfare at 3.63%, Capital Gains at 3.449%, and Master Bounty Hunter at 2.31%. This makes enough sense since they are by far the shortest CtM endings by number of fails and screens, so the odds of success are less extreme. The remaining values are as follows:

— Cleaned 'em Out at 0.997%
— Toppat Recruits/Pardoned Pals at 0.604% each
—Triple Threat at 0.128%
— Jewel Baron at 0.111%
— Toppat King and Free Man at 0.096%

In very very last are the Presumed Dead endings. In order of most to least likely, we have Toppat 4 Life, Little Nest Egg, and Valiant Hero, all at around 0.0001%. Rarest of all is Stickmin Space Resort, on account of having the second instance of a timeout leading to a success, at 0.000001%. The rarest possible timeline of Sneaky Escapist -> Unseen Burglar -> Pure Blooded Thief -> Presumed Dead -> Stickmin Space Resort accounts for only around 1 in every billion successful games. The odds of this getting this particular route without failing, taking into account all of the previous assumptions, are 1 in 13 sextillion against, around 85,000 times less likely to happen then winning two Powerball jackpots back to back.

And they said majoring in mathematics wasn't good for diddly-squat. Check and mate!