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[Community Question] Statistics: Probability of catching a Pokémon in an Escalation Battle

One of our user asked:

This question is inspired by the Escalation Battles in Pokémon Shuffle. There's a couple of other Pokémon-related questions on here, but they don't address this specific problem.

The way an Escalation Battle works is, the Nth time you beat it, you have an N% chance of catching the Pokemon. If you've already caught the Pokémon, you get items instead. When N=100, you're guaranteed to catch the Pokémon, but the chance of having not caught it by then must be vanishingly small.

I've competed in a few Escalation Battles, and I always seem to catch the Pokémon when 15 ≤ N ≤ 25. It's been years since I studied statistical probability at school, but this doesn't seem very intuitive to me. So I started wondering about the cumulative probability - how likely you are to have caught the Pokémon after N levels.

Is there a general formula to calculate the cumulative probability of having caught the Pokémon after N attempts? How many attempts will it take for the cumulative probability to exceed 50%?


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