Pokemon Learning League Options

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This is disturbing D:

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When a Poké Ball is thrown at a wild Pokémon, the game uses a formula based on the wild Pokémon's current health, any status effect it may have, and that Pokémon's catch rate, to determine the chances of catching that Pokémon. The formula is as follows:

Where

• HP_max is the number of hit points the Pokémon has at full health,
• HP_current is the number of hit points the Pokémon has at the moment,
• rate is the catch rate of the Pokémon,
• bonus_ball is the multiplier for the Poké Ball used, and
• bonus_status is the multiplier for any status ailment the Pokémon has (2 for sleep and freeze, 1.5 for paralyze, poison and burn).

Given this formula, the maximum value for a (if the Pokémon could have 0 HP) would be catch rate * bonus_ball * bonus_status. The minimum value for a (for a Pokémon with full health) would be 1/3 * catch rate.

If a is greater than or equal to 255, then the Pokémon is caught. If not, then calculate b as follows:

Then generate 4 random numbers between 0 and 65535, inclusive. If the random numbers are less than or equal to b, then the Pokémon is caught; otherwise the ball shakes n times, where n is the number of random numbers that are less than b. Note that b ≥ 65535 if a ≥ 255.

Therefore, the probability p of catching a Pokémon, given the values a and b calculated above, is:

The second expression for p may be expanded as follows:

Since (2¹⁶ - 1)⁴ ≈ 2⁶⁴, we can approximate p with the following expression:

The percentage error in this approximation approaches 0 as a approaches 255, and does not exceed 0.02%.

For a constant probability p, the probability P that a player can capture the Pokémon with no more than r tries is:

Note that this is the cumulative probability function for a geometric distribution. The expected value of r is 1/p, that is to say, on average, a Pokémon that can be caught with probability p will be caught with 1/p tries.

The inverse problem, the number of tries, r, needed to have a probability P of capturing a Pokémon is: