Catch rate
Each species of Pokémon has a catch rate inherent to all its members. The catch rates are out of a maximum of 255. When a Poké Ball is thrown at a wild Pokémon, the game uses a formula based on its current health, any status effect it may have, and that Pokémon's catch rate, to determine the chances of catching that Pokémon.
Approximately, the probability of catching a Pokémon starts with the species' catch rate divided by 255. Then it is multiplied with the following factors:
- The health of the Pokémon (relative to its full health), which penalizes up to ⅓×.
- The type of Poké Ball, which augments by some ×.
- The status of the wild Pokémon, i.e. 2× increase for sleep or freeze, and 1.5× for others.
The effects stack multiplicatively (i.e. a 2× and a 3× will combine to be 6×, not 5×). If the calculation ends up greater than 1, the wild Pokémon is assured to be caught.
Exact formula
The exact formula is as follows:
Where
- HPmax is the number of hit points the Pokémon has at full health,
- HPcurrent is the number of hit points the Pokémon has at the moment,
- rate is the catch rate of the Pokémon,
- bonusball is the multiplier for the Poké Ball used, and
- bonusstatus 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 × bonusball × bonusstatus. The minimum value for a (for a Pokémon with full health) would be ⅓ × 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:
Trivia
- If the HP of a Pokémon to be captured is reduced to 1, usually by way of False Swipe or Tricking a Focus Sash onto it and using a move that would normally OHKO, such as Fissure, then only 2 would be subtracted from the part of the formula in the brackets. This makes it so that Pokémon with higher HP stats (such as Chansey) are easier to capture when their HP is reduced, despite low catch rates compared to others.
See also
References
- ポケットモンスター情報センター 2号館: ボール