Using compound probabilities to estimate how much RNG is acceptable in a tournament setting

Since time immemorial Pokémon players have been asking themselves one question: Why do my Pokémon miss so many moves? The hit-rate always seems unreasonably low compared to the listed accuracy, and the old joke always feels a little too true: If it isn’t 100%, it’s 50%; it either hits or it doesn’t. Missing important moves can be detrimental to a tournament run, so some players choose to restrict themselves in their teambuilding by only using perfectly accurate moves, but there is a way to limit the amount of random number generation (RNG) you expose yourself to without having to renounce the less accurate but more powerful moves. So, let’s investigate why it seems like Pokémon miss more moves than they should, how this can impact a tournament run, and how to limit exposure to RNG while still using inaccurate moves. In this article I will also introduce a tool that can help you estimate how much you are limiting yourself as a player by relying on inaccurate moves to connect.

Why do moves miss more than they should?

Here’s the simple answer: They don’t.

It only feels like they do. This is because people tend to remember negative experiences more vividly than positive ones. This phenomenon is called the negativity bias. Players don’t remember the 9 times that Draco Meteor connected, but they sure remember the one time it didn’t.

Negativity bias can be very powerful; 2023 Charlotte Regional Champion Justin Carris was so convinced that Play Rough missed more than it should according to its 90% accuracy rate, that he set up an experiment to prove it. In a livestream he set out to click Play Rough a total of 1000 times to confirm his suspicion, that Play Rough had a lower accuracy than advertised. In the end, Justin’s experiment ended with a total of 897 hits out of 1000, making for an 89.7% hit-rate, remarkably close to the listed 90%, disproving his hypothesis. His perception of Play Rough’s accuracy had been skewed by negativity bias.

We also tend to misjudge the chance of two events happening in conjunction, which is called compound probability. Simply explained, the chance of a Hydro Pump connecting is 80%. Bleakwind Storm also has an accuracy of 80%, but it feels like it misses much more. This is because it checks the accuracy for two targets instead of one. The compound probability of Bleakwind Storm connecting on both targets is only 64% (0.80 * 0.80 = 0.64). While Bleakwind Storm and Hydro Pump have the same chance of hitting a single target, people expect Bleakwind Storm to connect more often than it actually should, because they use the move as if it had an 80% chance of hitting both opponents!

How does missing moves impact a tournament run?

There are some players, a famous example being five-time Regional Champion Riley Factura, who are so averse to missing moves, that they restrict themselves to using only 100% accurate moves when building teams. This philosophy can create teams that are mostly free of RNG, greatly reducing the game’s variability. In doing this however, they pay an opportunity cost by not being able to use certain powerful moves, or even entire Pokémon. It’s very difficult to calculate whether the benefit outweighs the cost here, but there is something else that we can calculate: How likely am I to lose a tournament just by missing my moves?

Let’s imagine a simplified scenario: You are playing a tournament with a team so good, that you are almost guaranteed to win every game against every other player in the tournament. The only way for you to lose is due to bad RNG. How likely are you to lose no more than two out of eleven Swiss rounds?

I am aware that this is a gross simplification, but it can be relevant to a real-life tournament, we’ll get to how exactly the results can be applied later.

Your win-condition could be any one of several more or less probable scenarios, where the likelihood of them happening is x. Some examples could be:

• Hitting a Draco Meteor (x = 90%)
• Double-connecting a Heat Wave (x = 81%)
• Double-connecting a Bleakwind Storm (x = 64%)

Let’s look at the calculation and then compute the results for these win-conditions. If you aren’t interested in the mathematics of how to calculate these probabilities, feel free to skip ahead to until you see a graph, there will be a TL;DR.

Probability of winning a Best of 3

To win a Best of 3, you must roll a positive outcome on this win-condition two out of three time. There’s three possible ways to do this:

1. Win Game 1 and Game 2: P(x) = x*x
2. Win Game 1, lose Game 2 and win Game 3: P(x) = x*(1-x)*x
3. Lose Game 1, win Game 2 and win Game 3: P(x) = (1-x)*x*x

Let’s translate this into a formula:

P_Bo3(x) = x² + x²*(1-x)*2

Probability of winning at least 9 out of 11 Swiss rounds

To make it to Top Cut at a medium-sized Regional Championship, you would have to win at least nine out of 11 Swiss rounds. Again, we need to consider three possibilities. The probability of these events is calculated off the probability of winning a Best of 3 P_Bo3, which is in turn calculated off the probability of the win-condition x.

1. Winning all Matches:
   P(x) = P_Bo3(x)¹¹
2. Winning 10 Matches and losing 1:
   P(x) = P_Bo3(x)¹⁰ * (1-P_Bo3(x)) * 11
3. Winning 9 Matches and losing 2:
   P(x) = P_Bo3(x)⁹ * (1-P_Bo3(x))² * 55

The multiplier 11 is due to there being 11 different individual Swiss rounds you can lose, and the multiplier 55 is due to there being 55 possible combinations of losing two Swiss rounds.

If we add these terms together, we get the total probability of losing no more than 2 Swiss rounds due to bad RNG:

P_TopCut(x) = P_Bo3(x)¹¹ + P_Bo3(x)¹⁰*(1-P_Bo3(x))*11 + P_Bo3(x)⁹*(1-P_Bo3(x))²*55

Chi-Yu’s Graph of Ruin

If you didn’t quite follow all of this, don’t worry. Here’s the TL;DR: We now have functions to calculate the probability of winning a Best of 3, P_Bo3(x), and winning enough Swiss rounds to make it into Top Cut, P_TopCut(x), purely by whether we meet our win-condition or not. Now we can plot these on a graph, which I have called “Chi-Yu’s Graph of Ruin”, after the Pokémon that misses more moves than any other.

On the horizontal axis we have the likelihood of meeting our win-condition, while the vertical axis shows the compound probability of that win-condition happening twice in a Best of 3 (blue line) and repeating for at least nine out of eleven Best of 3’s, letting you advance into Top Cut (orange line). The graph also shows the initial probability of the win-condition as a baseline (dotted line).

A few interesting observations can be made here:

First, upwards of 50%, where the blue and the dotted line cross, the probability of winning a Round is higher than the probability of winning an individual game. This means, that the Best of 3 system lessens the impact of RNG for probabilities over 50%.

Second, there is also a point at which the Swiss system lessens the impact of RNG, meaning that the probability of winning enough Swiss rounds is higher than the probability of the win-condition. This crossover point is far higher than 50% and changes depending on the number of Swiss rounds to be played. For eleven rounds it’s around 75%, where the orange and the dotted line cross.

Towards the edges of the graph, the likelihood of making Top Cut approaches 0 on the left and 1 on the right. This means, that for win-conditions with a probability upwards of 90%, the probability of losing more than two Rounds due to bad RNG is almost zero, while for win-conditions below 40% it’s almost a certainty.

Let’s look at the previously mentioned win-conditions as examples:

• Hitting a Draco Meteor: x = 90%
• Double-connecting a Heat Wave: x = 81%
• Double-connecting a Bleakwind Storm: x = 64%

While double-connecting a Bleakwind Storm still has a higher likelihood of winning you a Best of 3 (70.45%) than it does of happening in a single game (64%), the probability of it winning you nine or more Swiss rounds is already very low, only 32.45%.

Double-connecting a Heat Wave is already much more likely and the probability only improves over multiple Swiss rounds, ending up at 92.16% compared to the initial 81%.

Surprisingly, relying on a Draco Meteor to connect has an almost non-existent chance of costing you Top Cut at a tournament. The Best of 3 and Swiss systems eliminate almost any likelihood of a miss costing you the tournament.

How to apply the probability graph to a real-life setting

These probabilities are all well and good, but unfortunately there is a caveat: They only work in an idealized, theoretical scenario. In reality, no player will have a team without a single bad matchup and be able to play absolutely perfectly, so that these probabilities will actually apply to their likelihood of making it into Top Cut of a Regional. If a player is already struggling to make it into Top Cut, then any bad RNG could be enough to cost them that placement.

While this graph may not offer the absolute likelihood of a player making Top Cut, you can use it to estimate how heavily RNG is going to impact your run. If your win-condition is double-connecting Heat Wave, then you’re only utilizing 92.16% or your potential as a player, because the other 7.84% are lost due to bad RNG.

Here’s how you can apply this to the team that you are using right now:

• Take the accuracy of all the inaccurate moves on your team to the power of the number of times you need to rely on that move to hit during one battle.
• Multiply all the resulting probabilities with each other.

That is the probability of your win-condition. 

Let’s look at an example of a player known for using RNG-heavy teams.

Example: Joseph Ugarte at the 2025 Sacramento Regional Championships

At the 2025 Sacramento Regional Championships, Joe Ugarte used a team with a total of six inaccurate moves. This may seem like a recipe for disaster, but when watching him play, you can observe, that Joe limits the amount of RNG he exposes himself to in one battle. He plays in a way that he is not solely relying on hitting all his Sleep Powders, and he also doesn’t rely on Jumpluff hitting Leaf Storm to win battles. In most of his battles, there were two conditions that Joe had to meet to win the game:

• Hit at least one out of two Sleep Powders
• Hit one Overheat with Torkoal to pivot out

The probability of connecting at least 1 out of 2 Sleep Powders is 1 minus the probability of hitting 0, which is 1 minus the square of 1 minus the accuracy of the move.

 1 – (1 – 0.75)² = 1 – 0.25² = 93.75%

We then multiply that with the accuracy of Overheat, which is 90%.

(1 – 0.25²) * 0.9 = 84.38%

The probability of Joe’s win-condition is then 84.38%. If we plug this into our graph, we get:

• A probability of success in a Best of 3 of 93.44%
• A probability of success in at least nine out of eleven Swiss rounds of 96.87%

We can see that, while Joe has a lot of inaccurate moves on his team, he actively limits his exposure to RNG by using them a limited amount or by allowing room for error. It’s a masterclass in calculated risk.

However, even this type of calculated risk can still backfire. This could be seen in Game 3 of the Finals, where Joe’s Torkoal ended up missing three Overheats in a row, an unlucky streak with a measly 0.1% chance of occurring.

Conclusion

Missing moves in Pokémon battles is a frustrating experience. It can often feel like your Pokémon miss more moves than they should. There are two reasons for this: Negativity bias, which is when we remember negative experiences more vividly than positive ones, and overestimating compound probabilities. The chance of a spread move double-connecting is lower than most people would think.

In a tournament run, the Best of 3 and Swiss systems can lessen the impact of low-probability events, because they allow more room for error than single elimination Best of 1. If you have a win-condition with a certain probability, the likelihood of meeting that win condition enough to win a Best of 3 or nine out of eleven Swiss rounds can be calculated with the following equations: 

P_Bo3(x) = x² + x²*(1-x)*2

P_TopCut(x) = P_Bo3(x)¹¹ + P_Bo3(x)¹⁰*(1-P_Bo3(x))*11 + P_Bo3(x)⁹*(1-P_Bo3(x))²*55

These can be plotted in Chi-Yu’s Graph of Ruin, which shows that win-conditions with a higher likelihood than 80% still have a chance higher than 90% of being met in at least nine out of eleven Swiss rounds. 

These probabilities cannot be used directly as the likelihood of winning a Best of 3 or making Top Cut, but should rather be seen as the factor by which you are limiting yourself by relying on these RNG-dependent win-conditions.

Even when playing a team with many inaccurate moves, you can reduce your exposure to bad RNG by limiting the number of times you use these moves, and by allowing room for error, such as only needing to hit one out of two Sleep Powders, instead of one out of one.

I hope you enjoyed reading this article, as it has been the most fun to write of all my articles so far. Please let me know whether the topic was presented in a way that was relatively easy to understand, as I’m a bit worried i didn’t do a good enough job explaining the calculations. It will be a little while before i can write another article, since i have to prepare for the Sevilla Special Event and lock in on my studies. In the meantime, feel free to let me know what you like and dislike about my articles and if there are any topics you would like to see me cover in the future. Also, bonus points for anyone who can recognise what the drawing on the blackboard in the first picture of the article are about.