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BulbapediaEach species of Pokémon has a catch rate, a number out of a maximum of 255, that applies to all its members. 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 by the following factors:
• The health of the Pokémon (relative to its full health), which can penalize up to ⅓× at full health.
• The type of Poké Ball, which can augment 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 — for example, a 2× and a 3× will combine to be 6×. If the calculation ends up greater than 1, the wild Pokémon is assured to be caught.
The rarity of the Pokémon (i.e. how often it appears in the wild) is totally independent from its catch rate after it is encountered. Even rare Pokémon may have high catch rates, making them relatively easy to catch in battle, such as Nosepass (catch rate 255).
The exact formula is as follows:
a=
(3 x HPmax - 2 x HPcurrent x rate x ballbonus x ball
status3 x HP
MaxWhere
• 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.
Calculate b as follows:
b=1048560 / The square root of the square root of 16711680 over a = (2 to the sixteenth power - 1) x the second root of a over the fourth root of two to the eighth - 1 = (Two to the sixteenth - 1) x the fourth root of a over two to the eighth - 1
Then generate 4 random numbers between 0 and 65535, inclusive. For each number, check whether it's less than b; if so, the ball shakes once. If any number is greater than or equal to b, the Pokémon immediately breaks free. If all four of the numbers are less than b, then the Pokémon is caught. Note that if a is 255 or greater, then b is 65535 or greater, and then the Pokémon is guaranteed to be caught.
Therefore, the probability p of catching a Pokémon, given the values a and b calculated above, is:
p= 1 : a
> 255
(b + 1 over 2 to the sixteenth) to the first : a < 255
The second expression for p may be expanded as follows:
p= 1 over 2 to the sixty first power ( (2 to the sixteenth - 1) the forth root of a over 2 to the eighth - 1 + 1)) to the first = (2 to the sixteenth -1 over 2 to the sixty first x a over 2 to the 8 -1 + ..
Since (216 - 1)4 ≈ 264, we can approximate p with the following expression:
p is approximately equal to a over 2 to the 8 -1
The percentage error in this approximation approaches 0 as a approaches 255, and does not exceed 21.2%.
For a constant probability p, the probability P that a player can capture the Pokémon with no more than r tries is:
P= = 1 - [1 - p] to the power of p
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:
r= log (1 - P) over the log of (1 - p)
