Y by GA34-261
(Thanks Random Stranger for the idea and I will be making it so it is extremely specific to your solution.)
We are playing Pokemon Scarlet and Violet and you are fighting a friend. You and your friend don't have any items at all and the pokemon does not have any held items. Your friend challenges you to a battle because he just said nah I'd win.
You two start the battle using only one Pokemon each which neither of you knows the type of the other. Luckily he had used a level 28 Squirtle and you had used a level 25 Pikachu. Surprisingly both of the Pokemon each have one HP. Your Pikachu has a move set of one single move of Thunder with 10/10 PP and has 1 HP because you forgot to go to the Pokemon center. Your Pikachu also has a bad IV stat in speed with 1/15 and the 252 EV speed stat of the Squirtle combined with a perfect IV stat in speed makes it so it guarantees to always out speed your move. To account for that he made his Squirtle have 1 HP on purpose for absolutely no reason.
After he saw what kind of moves you have and since that person was so cocky and confident that they decided to gamble all their moves with each having an equal chance of being used. Their Squirtle has a move set of Protect 10 PP which has 100% chance of being used and has has the success probability multiplied by 1/3 every time it is being used (Meaning the second time it is being used has a 33% chance of succeeding and a third time it will be 11%. This also ignores the rules of how the move is regularly used by making the 4th move 1/27 instead of it being a guaranteed fail and so on.), Tackle which has a 100% chance to hit having 10/35 PP , Water Gun which has a 100% chance of hitting with 10/25 and Rain Dance with 5/5 PP and 100% chance of being used. If the amount of PP reaches 0 it will be unavailable for the rest of the fight meaning that the probability for each other move to be used goes from 25% all the way to 33%.
For everyone who wants to solve the easy part. If the probability that Squirtle will survive turn 1 when simplified is a/b then what is a+b?
Alright so for Squirtle to survive turn one then we try to find out how Squirte will faint at turn one. First of all Pikachu needs to hit Thunder bolt at a 70% chance then get a 25% chance that the Squirtle will use Rain Dance so that the Squirtle will not faint the Pikachu because it didn't attack. Another possible option is for it to choose Water Gun and miss so it would be a 70% * 25% * 5% chance for Pikachu to faint Squirtle. 70% = \frac{7}{10}, 25% = \frac{1}{4}, and 5% = \frac{1}{20} So the complement of what we are trying to find is
. The complement of this would be 
or 
. The final thing we can do is to make sure it is simplified and add the numerator and the denominator which is 
and 
so 
This should be final answer. (Mathdash rating 800)
For the very hard question, What is the probability that the Squirtle will win this fight? (This is going to be a very long arithmetic series with a lot of cases. The max amount of turns this fight can have is 11 turns.)
We are playing Pokemon Scarlet and Violet and you are fighting a friend. You and your friend don't have any items at all and the pokemon does not have any held items. Your friend challenges you to a battle because he just said nah I'd win.
You two start the battle using only one Pokemon each which neither of you knows the type of the other. Luckily he had used a level 28 Squirtle and you had used a level 25 Pikachu. Surprisingly both of the Pokemon each have one HP. Your Pikachu has a move set of one single move of Thunder with 10/10 PP and has 1 HP because you forgot to go to the Pokemon center. Your Pikachu also has a bad IV stat in speed with 1/15 and the 252 EV speed stat of the Squirtle combined with a perfect IV stat in speed makes it so it guarantees to always out speed your move. To account for that he made his Squirtle have 1 HP on purpose for absolutely no reason.
After he saw what kind of moves you have and since that person was so cocky and confident that they decided to gamble all their moves with each having an equal chance of being used. Their Squirtle has a move set of Protect 10 PP which has 100% chance of being used and has has the success probability multiplied by 1/3 every time it is being used (Meaning the second time it is being used has a 33% chance of succeeding and a third time it will be 11%. This also ignores the rules of how the move is regularly used by making the 4th move 1/27 instead of it being a guaranteed fail and so on.), Tackle which has a 100% chance to hit having 10/35 PP , Water Gun which has a 100% chance of hitting with 10/25 and Rain Dance with 5/5 PP and 100% chance of being used. If the amount of PP reaches 0 it will be unavailable for the rest of the fight meaning that the probability for each other move to be used goes from 25% all the way to 33%.
For everyone who wants to solve the easy part. If the probability that Squirtle will survive turn 1 when simplified is a/b then what is a+b?
Alright so for Squirtle to survive turn one then we try to find out how Squirte will faint at turn one. First of all Pikachu needs to hit Thunder bolt at a 70% chance then get a 25% chance that the Squirtle will use Rain Dance so that the Squirtle will not faint the Pikachu because it didn't attack. Another possible option is for it to choose Water Gun and miss so it would be a 70% * 25% * 5% chance for Pikachu to faint Squirtle. 70% = \frac{7}{10}, 25% = \frac{1}{4}, and 5% = \frac{1}{20} So the complement of what we are trying to find is
. The complement of this would be 
or 
. The final thing we can do is to make sure it is simplified and add the numerator and the denominator which is 
and 
so 
This should be final answer. (Mathdash rating 800)For the very hard question, What is the probability that the Squirtle will win this fight? (This is going to be a very long arithmetic series with a lot of cases. The max amount of turns this fight can have is 11 turns.)
This post has been edited 6 times. Last edited by PikaVee, Jun 1, 2025, 2:27 PM
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