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# Taking Care of the Pennies – Small Probabilities in Pokémon

Today we’re going to be taking a closer look at small probabilities in Pokémon – how they can stack up and what they really mean. I’m going to be using the game that inspired this analysis as a reference point; the third game of the North American International Championship Masters’ division finals between Paul Chua and Christopher Kan, particularly the final few turns. After this game, a lot of people were confused about which of the players had received more good fortune during the game, and understandably so. Though the probabilities of a single Rock Slide flinch or full paralysis are relatively slim, it’s easy to lose track of how they can accumulate. The match in question is linked below, with the final game beginning at the 29:30 minute mark.

Since the main focus of this article is the final few turns of the game, I will present a brief discussion of the first few turns in order to set the scene for the endgame. If you’re only interested in the analysis of the low-probability outcomes, I recommend jumping ahead to turn 6. Before we get into it, here’s a quick disclaimer: This analysis was not conducted with any intention of implying that either player “deserved” the win more than the other, and is merely intended to illustrate the probabilities associated with repetitions of low-probability outcomes and the fluctuations therein. The probabilities were only calculated for the final few turns, and don’t take into account the earlier turns, nor the possible variations in each player’s moves, which could potentially be hugely influential.

() vs ()

Turn 1:

Chris leads Arcanine and Porygon2 into Paul’s Snorlax and Tapu Koko. Download boosts Porygon2’s Attack, whilst Intimidate lowers Snorlax’s. As Paul has already revealed Electrium Z on his Tapu Koko in game 1, Chris knows that it threatens the KO on his Arcanine. Though Chris’s Arcanine was in fact bulky enough to survive a Z-Thunder into its Protect plus a Thunderbolt on the following turn, if Paul predicted this and opted not to use his Z-move on turn 1, Chris would be left in an extremely awkward position. Chris knows that landing a Toxic on Snorlax is his most promising way of beating it, so risking losing Arcanine this turn is not something he can afford to do. He switches it out to Tapu Koko, which narrowly avoids being KO’d by Gigavolt Havoc (notably, Tapu Koko is guaranteed to survive this attack). Porygon2’s boosted Return takes away about a third of Snorlax’s health, and Snorlax proceeds to set up with Curse.

() vs ()

Turn 2:

As Chris’s Tapu Koko is shiny, Paul must be aware that it must be Timid (as shiny Tapu Koko is only available through a nature-locked event). As it is holding Electrium Z (which Paul has confirmed he was aware of prior to this turn), it threatens a KO on Paul’s Snorlax, as illustrated by the damage calculation below. Since Snorlax is a key component to his game plan, some players in Paul’s position would be extremely tempted to switch it out to preserve it, rather than risk losing it should he lose the speed tie. However, Paul decides that his odds of winning the game if he switches Snorlax out in this position are lower than the odds that he will win the speed-tie, and decides to keep Snorlax in. Paul’s Tapu Koko wins the speed-tie, KOing its opposite number with Dazzling Gleam. Porygon2 deals around 65% damage to Koko with Return, and Snorlax sets up a second Curse.

Z-Thunderbolt vs Snorlax

Would Paul have lost too much momentum by switching out his Snorlax in this position? Switching in Garchomp would have negated the damage from the potential Gigavolt Havoc, but would have left Garchomp at -1 Attack after Arcanine switched back in, staring down a near-full health Porygon2. His other option would have been to switch in Ninetales, which would have been brought down to its Focus Sash, ripe to be picked off by Arcanine’s Extreme Speed on the following turn. It’s easy to criticise the decision to keep Snorlax on the field in this position as needlessly risky, but it’s clear that Paul’s alternative options were likely to put him in a worse position even if he were to win the speed-tie. On the flip-side of this, had Chris allowed his Arcanine to be KO’d by Gigavolt Havoc on turn 1, he would have been able to bring in Tapu Koko without taking damage, able to threaten the KO on Snorlax after the damage from Porygon2’s Return. However, with Arcanine out of the picture, the decision to switch Snorlax out would become considerably more attractive for Paul, as he would be able to set it up later without the fear of taking Toxic damage.

() vs ()

Turn 3:

Chris’s Arcanine returns to the field, setting Snorlax’s attack back to neutral. It now threatens the possible KO on Tapu Koko with Extreme Speed (an 81% roll as shown by the damage calculation below, though the players may not have been aware of this), but landing Toxic on Snorlax is also a priority, as the extra turn is likely to pay dividends once the damage starts to accumulate. Paul opts to Protect with Tapu Koko, as Chris does indeed prioritise landing Toxic on Snorlax over denying Tapu Koko a final attack. His Porygon2 attacks into Tapu Koko’s Protect, as Snorlax sets up yet another Curse, bringing it to +1 Attack and +3 Defence.

Chris's Arcanine's Extreme Speed vs Tapu Koko

() vs ()

Turn 4:

With Tapu Koko’s Protect used last turn, it is now vulnerable to Arcanine’s Extreme Speed. However, at +1 attack, Snorlax now threatens the KO on Arcanine with its boosted Facade. Chris, therefore, expects Snorlax to attack into the Arcanine slot, and Tapu Koko to attack into the Porygon2 slot in case Arcanine decides to Protect instead. Chris decides that his best option is to Protect with Arcanine and Recover with Porygon2, allowing Toxic damage to slowly build up on Snorlax. Paul makes a good read in response, doubling into the Porygon2 with Thunderbolt and Facade, leaving it with around 40% of its HP remaining. The poison damage then activates Snorlax’s Figy Berry, which returns it to near full HP.

After the disastrous loss of the Tapu Koko speed-tie and now this solid turn from Paul, things look dire for Chris. Porygon2 is now in range of Thunderbolt + Facade even if it uses Recover, and losing it may leave Chris short of answers to Paul’s Garchomp.

() vs ()

Turn 5:

Chris once again threatens the possible KO on Tapu Koko with Extreme Speed, and could save his Porygon2 from fainting this turn by going for Extreme Speed and Recover. However, doing so may leave Porygon2 in range of a Blizzard from Ninetales or even a Rock Slide from Garchomp, and with Snorlax’s HP as high as it is, it would likely be able to stick around for 3 more turns after this one should he fail to damage it further. As a result, Chris decides to double into the Snorlax with Flare Blitz and Return (note – though Ice Beam here would technically do more damage despite the Download boost, the difference would be negligible. Conversely, if Porygon2 lands a critical hit, Return would deal considerably more damage than Ice Beam, making Return arguably the better option). Paul does indeed double into the Porygon2, which faints to the combination of attacks. After Toxic damage and the attacks from both Porygon2 and Arcanine, Snorlax is left with just below half of its HP, enough to allow it to survive two more turns of Toxic damage. As a result, using Recycle would likely be relatively futile, and Paul is better off instead going on the offensive for the remaining two turns of Snorlax’s life.

vs ()

Turn 6:

Chris now stares down a 4-2 Pokémon deficit, but with Paul’s Snorlax almost off the field and his own Snorlax still at full health, the game is far from over. For the fourth turn running, Arcanine threatens the potential KO on Tapu Koko with Extreme Speed. Paul finally expects Chris to take the bait and switches Tapu Koko out to preserve it perhaps for when Arcanine is no longer of the field. Chris makes an excellent play in response, hitting the switching in Garchomp with Toxic. By doing so, he not only avoids taking damage from Rough Skin, but also sets Garchomp’s Toxic counter 1 turn ahead. Chris’s Snorlax uses Curse, reducing the damage of the incoming Facade, which activates Snorlax’s Aguav Berry, restoring it to near full HP.

Despite the dominant, aggressive start from Paul, the remainder of the game will play out in such a way that in order to win, Paul will need to prevent Chris’s Snorlax from moving at least once in order to take it down in time. He will need to rely on Rock Slide flinch chances and Paralysis and Freeze chances from Thunderbolt and Blizzard respectively. In the interest of examining exactly how these probabilities line up over the course of each turn, I’ll be placing win percentage odds before each of the remaining turns. This will represent the odds of each player winning, calculated using a couple of key assumptions:

1. The players make the moves that they do. This does not mean that I necessarily agree that each player’s moves are entirely optimal, but using this assumption allows the odds to be calculable, if only in retrospect.

2. Chris will lose if his Snorlax is unable to move on any turn from 7-10, OR if it is unable to move on both turns 11 AND 12. This is a (mostly) fair assumption given how the game eventually unfolds, though failing to factor in missed damage from potential missed Rock Slides will make the final estimates slightly biased in Paul’s favour.

The odds won’t be exact as the entire set of possible outcomes would be impossible to account for, but will factor in flinch chances from Rock Slide, Paralysis & Full Paralysis chances from Thunderbolt, and Freeze chances from Blizzard. They will also factor in the various possible outcomes of turn 9, which I will explore in more detail later on.

vs ()

Turn 7:

Based on our assumptions, Chris’s Snorlax has around a 26% chance of being able to clutch out this game going into this turn. Garchomp locks into Rock Slide as expected, which flinches the Arcanine. Snorlax uses Recycle, as Paul’s Snorlax hits it with another Facade, which leaves it at around 80% health after it consumes its newly recycled Aguav Berry. Paul’s Snorlax now faints to the Toxic damage. The flinch on the Arcanine this turn is largely irrelevant, and may, in fact, benefit Chris, as it prevents him from taking unnecessary recoil damage from attacking.

vs ()

Turn 8:

Snorlax still has its back to the wall, with only a 36% chance of being able to stay mobile for the requisite number of turns. Tapu Koko switches in for the fainted Snorlax. Arcanine still threatens the possible KO with Extreme Speed, but Chris instead opts to use Flare Blitz, which not only allows Tapu Koko to get off a final attack off before it is KO’d, but also risks a few further punishes. Firstly, it risks Arcanine being flinched by Rock Slide, and, therefore, failing to KO the Tapu Koko. Secondly, if Rock Slide misses the Arcanine, or if Garchomp switches out, the Mago Berry will not activate, and Tapu Koko’s Thunderbolt would likely be able to pick up the KO from this range. Finally, it risks Paul opting to use Thunder on the Arcanine, which would have a good chance of KOing it even after the Mago Berry activation. Despite this, Chris is not confident in his Arcanine’s ability to pick up the KO with Extreme Speed, though in reality, we know that Arcanine had better odds of KOing Tapu Koko with Extreme Speed than it did of being able to move through Rock Slide’s flinch chance.

Chris’s underestimation of Extreme Speed’s damage output is punished here, as it allows Tapu Koko to get a Thunderbolt off onto Snorlax, which deals solid damage and snags a paralysis. Arcanine does manage to move, however, KOing Tapu Koko with Flare Blitz, as does Snorlax, recycling and eating its Aguav Berry.

vs

Turn 9:

The Paralysis is detrimental to Chris’s chances, but they have improved nonetheless as Snorlax was able to make it through the previous turn. Ninetales switches in for the fainted Tapu Koko. The combination of Rock Slide and Blizzard fail to pick up the KO on Arcanine (despite the arguably needless recoil damage it took by using Flare Blitz as opposed to Extreme Speed last turn), which hits back with Flare Blitz, bringing Ninetales down to its Focus Sash and fainting to recoil damage in the process. Snorlax is able to move despite the flinch and paralysis chances, recycling its berry once again.

This turn plays itself, with the exception of one Pokémon – Arcanine. Arcanine has 2 main options this turn; Protect or Flare Blitz, and each has its merits. Had Chris used Protect this turn, it would have ensured that Snorlax took spread damage from the Rock Slide and Blizzard on the following turn, reducing their damage. However, the chip damage from hail would likely have put Arcanine in range of the Rock Slide and Blizzard on the following turn. This is negligible, however, as Arcanine only really needs to break Ninetales’s Focus Sash, which it can do for certain with Extreme Speed on the following turn, as Paul’s Ninetales does not carry Protect. This also carries the benefit of not risking Arcanine being flinched by Rock Slide, or taking a critical hit from either attack and fainting before it can use Flare Blitz. However, the cost of failing to break Ninetales’s Focus Sash with Arcanine is relatively small in terms of the eventual probabilities of victory. On the other hand, Flare Blitz carries with it a number of benefits to counterbalance the aforementioned risks. Firstly, if Arcanine avoids Garchomp’s Rock Slide, it will be able to pick off the Ninetales on the following turn with Extreme Speed, allowing Chris to win the game regardless of whether or not Snorlax is able to move this turn. Additionally, if Arcanine is hit by Rock Slide and avoids flinching, it will have a 10% chance of burning Ninetales, KOing it through its Focus Sash, again giving Chris the win regardless of all other outcomes.

The optimal option in this specific scenario depends on a number of factors, not all of which can realistically be accurately accounted for. If we assume that Arcanine will survive the combination of attacks unless either of them lands a critical hit and that the reduction in damage to Snorlax caused by making the attacks on the following turn spread damage would be negligible in the long-run, then the possible favourable RNG-related outcomes brought about by using Flare Blitz would outweigh the potential negative outcomes such that it would produce a win rate as much as 1.3 times higher than that of the “safer” option of opting to use Protect followed by Extreme Speed. Though this particular situation is too complex for us to say with surety which option is better, it does help illustrate an important point: It is not always correct to pursue the option with the least risk – there are some situations in which the positive low-probability outcomes of a given option are impactful enough to allow them to outweigh the associated risks to the point that the ‘riskier’ option actually produces a higher win rate. The inverse of this is also true, and being able to identify where and when each option will lead to the highest percentage chance of victory is crucial to any player who wants to maximise their win rate.

vs

Turn 10:

Though Arcanine failed to produce either of the favourable outcomes that would win the game for Chris on the spot, Snorlax has stuck around for long enough to finally swing the odds in its favour, but still has one major obstacle remaining in the form of turn 10. The single-target damage from Garchomp’s Rock Slide and Ninetales’s Blizzard combine to deal enough damage to indicate that the hail damage at the end of the turn plus another Blizzard on the following turn would be enough to pick up the KO. However, Snorlax is once again able to get its Recycle off, restoring itself back up to a healthy level of HP. Garchomp faints to the Toxic damage, leaving Snorlax 1v1 against the Ninetales.

vs

Turn 11:

Snorlax has overcome tough odds to get to this position, but it still isn’t quite out of the woods yet. Two single target Blizzards plus hail damage should fail to KO Snorlax from this range, so it would have to be fully paralysed on both this turn and the following turn for Ninetales to be able to KO it. It is able to move through paralysis, picking up the KO on Ninetales with Facade and sealing the game and the match in Chris Kan’s favour.

#### Some final words

Over-reliance on secondary effects is something that is generally frowned upon in competitive Pokémon, however, it is important to acknowledge the role they play in many games, particularly in situations like the one above in which the secondary effects were given multiple opportunities to occur. Given how the game played out, Paul’s chances of winning the game based on secondary effects alone were as high as 74% going into turn 7. While these figures are skewed by Rock Slide’s unusually high secondary-effect probability of 30%, even if we were to lower Rock Slide’s flinch chance to 10% Paul’s chance of winning based on secondary effects alone would still be as high as 48% going into turn 7, and if Rock Slide had no flinch chance, Paul’s win percentage would be around 24% based on Thunderbolt Paralysis and Blizzard Freeze chances alone.

Low-probability secondary effects are easy to complain about when they occur, however, their singular occurrence often receives an undue level of animosity due to the tendency of many players to underestimate the accumulative effect of their repeated use. Analysing and understanding this effect can help players to make better mental estimates of probabilities, allowing them to make more informed decisions about which lines of play give them the best odds of winning a game.