Catch rate

Catch rate refers to the probability of catching a wild Pokémon in the core series. The catch rate not only concerns the Pokémon species, but also their HP, the type of Poké Ball used by the player, or even their status condition. If the player is unsuccessful at catching a Pokémon for too long, the Pokémon may flee.
The lowest catch rate is 3, which belongs to the Beldum family and almost every legendary Pokémon in the game. A low catch rate means that the Pokémon is very hard to catch, often meaning the player must use over 20 Poké Balls or more. The highest catch rate is 255, which belongs to small Pokémon such as Caterpie or Pidgey, which can be easily caught with a Poké Ball without the need to damage them. A Master Ball also has a 100 percent chance of catching a Pokémon.
==Formulae==
The formulae for catch rate is more complex and different in each generation.
===Generation I===
In Generation I, the capture mode differs from the later generations. To determine the catch rate, there are important steps performed below. If, at any point, the Pokémon breaks free, the steps will never be performed.
:1. If Master Ball is used, the Pokémon is always caught, no matter what.
:2. Generate a random catch rate number, denoted by Θ (Theta), depending on the type of ball used.
*Poké Ball:
\Theta_{PokeBall} = rand(255)
*Great Ball:
\Theta_{GreatBall} = rand(200)
*Ultra Ball and Safari Ball:
\Theta_{UltraAndSafariBall} = rand(150)
:3. The Pokémon is caught if...
*If it is asleep or frozen and Θ is less than 25.
*It is paralyzed, burned, or poisoned and Θ is less than 12.
:4. Otherwise, if Θ minus the status threshold (above) is greater than the Pokémon's catch rate, the Pokémon breaks free. 
:5. If not, generate a random value (Denoted by the letter M).
M = rand(255)
:6. Calculate Λ
Formula:
\Lambda = \frac{\left ( \beta_{max} \times 255 \times 4\right )}{(\beta_{current} \times B)}
:β_max - The maximum HP of a Pokémon
:β_current - The current HP of a Pokémon
:B - The Poké Ball's integer
The minimum value of Λ is 1 and its maximum value is 255. The value of B is 8 if a Great Ball is used or 12 otherwise.
7. If Θ is greater than or equal to Λ, the Pokémon is caught. Otherwise, the Pokémon breaks free. In practical terms, lowering the target's HP to 1/3 of its maximum will guarantee capture with a Poké Ball, while lowering it to 1/2 will guarantee capture with a Great Ball.
If the Pokémon broke free, the steps below are performed to determine how many times the ball will shake.
:1. Calculate δ (shake probability):
\delta = \frac {\Omega \times 100}{B}
Where the value of Ball is 255 for the Poké Ball, 200 for the Great Ball, or 150 for other balls.
:2. If δ is greater than or equal to 256, the ball shakes three times before the Pokémon breaks free.
:3. If not, use this formula.
x = \frac{\delta \times \Lambda}{255 + \rho}
Where ρ is the status modifier. It is 10 if the Pokémon is asleep or frozen or 5 if it is paralyzed, poisoned, or burned.
4. If...
*x < 10: the Ball misses the Pokémon completely.
*x < 30: the Ball shakes once before the Pokémon breaks free.
*x < 70: the Ball shakes twice before the Pokémon breaks free.
*Otherwise, the Ball shakes three times before the Pokémon breaks free.
=====Approximated Probability=====
Using a simulation of the capture algorithm, a more straightforward and simple formula for the probability of catching a Pokémon was found.
Probability = \mu_{0} + \mu_{1}
μ_0:
\mu_0 = \frac{\rho_{ailment}}{B_{mod} + 1} 
Where:
:ρ_ailment = 12 if poisoned, burned, or paralyzed, 25 if frozen or asleep, 0 otherwise.
:B_mod = 255 if using a Poké Ball, 200 if using a Great Ball, and 150 otherwise.
μ_1:
\mu_1 = \left ( \frac {(\Omega + 1)}{(B_{mod} + 1)} \right ) \times \left ( \frac {(\Lambda + 1)}{256} \right )
*Lambda is defined in the above section for the capture method.
*Omega (catch rate) (given as an integer value) is stated on each individual Pokémon's article.
==General Capture Method (Generation II onwards)==
The capture algorithms in Generation II and onwards have three crucial parameters: the modified catch rate, the "shake probability", and the "shake checks". Generation II's shake checks handle differently than further generations.
=====Modified Catch Rate=====
The modified catch rate, Ω (Omega), is the catch rate after various factors such as weakening the Pokémon and using stronger Poké Balls are taken into consideration. A modified catch rate may never fall to 0 (that is, render a Pokémon impossible to capture), but it may cause the modified rate to fall below its original unmodified catch rate (such as from high health, Heavy Balls, Baiting in the Safari Zone, or the dark grass penalty in Generation V). In Generation III and Generation IV, the modified catch rate may never fall below 1.
=====Shake probability=====
The shake probability, φ (phi), is a value that determines the probability that a single shake check passes..
=====Shake checks=====
Shake checks are performed to determine whether the Pokémon will be caught or, if the Pokémon breaks free, the number of shakes that will occur before it does so.
In Generation II, whether a Pokémon will be caught is determined before any shake checks are performed, and shake checks are only performed if the Pokémon is not caught.
===Capture Method (Generation II)===
=====Formula=====
The modified catch rate Ω (Omega) is calculated in Generation II as follows:
 \Omega = \frac {(3 \times \beta_{max} - 2 \times \beta_{current}) \times R}{\left ((3 \times \beta_{max}), 1 \right ) + \rho_{status}}
with the final value rounded down to the nearest integer, where
:β_max - The maximum HP of the targeted Pokémon
:β_current - The current HP of the targeted Pokémon
:ρ_status - The modifier for any status condition the Pokémon has (10 for sleep or freeze, 0 otherwise). It was intended to equal 5 for paralyze, poison, or burn, but due to a glitch, the game skips this check.
:R - The catch rate of a Pokémon.
*If 3 × β_max > 255, then both 3 × β_max and 2 × β_current are halved twice (and rounded down after each division) for use in the formula, as the values used are unsigned 8-bit integers. If the latter product is 0, it is set to 1 instead.
*If the Pokémon's HP is 342 or greater, the 3 × β_max value will be truncated and the subtraction may underflow, giving bizarre results and even making it possible for the game to freeze; however, no such Pokémon can be legitimately encountered in-game.
*Ω is capped at 255.
=====Shake probability=====
The shake probability φ is determined from the table below, depending on the value of Ω.
Ω	0-1	2 	3 	4 	5 	6-7	8-10	11-15	16-20	21-30	31-40	41-50	51-60	61-80	81-100	101-120	121-140	141-160	161-180	181-200	201-220	221-240	241-254	255
φ	63 	75	84	90	95	103	113 	126 	134 	149 	160 	169 	177 	191 	201 	211 	220 	227 	234 	240 	246 	251 	253 	255
=====Shake checks=====
First, a check is performed to determine whether the Pokémon is caught at all. A random number between 0 and 255 is generated, and if this number is less than or equal to a, the Pokémon is caught.
Shake checks are only performed if the Pokémon is not caught. A single shake check consists of generating a random number between 0 and 255 and comparing it to b. This is done at most three times, but if the number generated in a given shake check is greater than or equal to b, no further shake checks will be performed. The number of times the ball shakes is the same as the number of shake checks that were performed.
==Capture Method (Generation III-IV)==
===Modified Catch Rate===
The modified catch rate, Ω, is calculated in Generation III and Generation IV as follows:
\Omega = \frac {(3\times \beta_{max}-2\times \beta_{current}) \times R \times \xi_{ball}}{3 \times \beta_{max}}\times \xi_{status}
Where:
*β_max is the number of hit points the Pokémon has at full health,
*β_current is the number of hit points the Pokémon has at the moment,
*R is the catch rate of the Pokémon (which may be modified due to use of apricorn balls or actions in the Safari Zone),
*ξ_ball is the multiplier for the Poké Ball used, and
*ξ_status is the multiplier for any status condition the Pokémon has (2 for sleep and freeze, 1.5 for paralyze, poison, or burn, and 1 otherwise).
The formula for R (the catch rates are denoted in η):
R = (\eta_1 \times n_1)+(\eta_2 \times n_2)...
equal to the summation of:
R_{total} = \sum (\eta_{catch} \times n_{pokeballs})
If a Pokémon could have 0 HP, the maximum value for a would be R × rho_ball × rho_status. For a Pokémon with full health and no status condition, and with a neutral ball used, the minimum value for Omega would be R/3.
The formula is slightly different when applied to the Apricorn balls in HeartGold and SoulSilver. The modifiers for these balls are applied directly to the Pokémon's catch rate, rather than in the formula. In this case, bonusball is always 1, and the catch rate cannot go higher than 255. This means that for Pokémon whose catch rate is already 255, such as Rattata, the Apricorn balls do not make the capture any more likely than a regular Poké Ball.
=====Shake Probability=====
The shake probability φ is calculated as follows:
\varphi = \frac{1045860}{\sqrt{\sqrt{\frac{16711680}{\Omega}}}}
The divisions and square roots all round down to the nearest integer.
===Shake checks===
To perform a shake check, a random number between 0 and 65535 (inclusive) is generated and compared to b. If the number is greater than or equal to b, the check "fails".
Four shake checks are performed. The Pokémon is caught if all four shake checks succeed. Otherwise, the Poké Ball will shake as many times as there were successful shake checks before the Pokémon breaks free.
If Omega is 255 or greater, the capture will always succeed and no shake checks will be performed.
===Probability of capture===
The probability "mu" of catching a Pokémon, given the values Omega and Phi calculated above, approximates φ/255. However, due to rounding errors produced when calculating φ, this approximation can be significantly inaccurate: all a values greater than 200, for instance, yield the same φ value, 65535 (which results in a 99.994% chance of a successful capture).
For a constant probability mu, the probability P that a player can capture the Pokémon with no more than tau tries is:
P = 1 - e^{\tau \ln(-\mu+1)}
Note that this is the cumulative probability function for a geometric distribution. The expected value of tau is 1/mu, that is to say, on average, a Pokémon that can be caught with probability mu will be caught with 1/mu tries.
The inverse problem, the number of tries, tau, needed to have a probability P of capturing a Pokémon is:
 \tau==\frac{\ln(1-P)}{\ln(1-\mu)}
==Capture Method (Generation V)==
===Modified Catch Rate===
Generation V follows the formula in Generation III-IV, with all divisions above rounded down to the nearest multiple of 1/4096. However, there are three alterations:
* rhostatus is now 2.5 for sleep and freeze (instead of 2).
* Capture Power factor: if a Capture Power is active, the catch rate is multiplied by a factor and rounded down to the nearest multiple of 1/4096.
** Capture Power ↑: ×1.1
** Capture Power ↑↑: ×1.2
** Capture Power ↑↑↑, S, or MAX: ×1.3
* Dark grass factor: This factor is multiplied into the HP factor:
\Omega_{DarkGrass} = 3\times \beta_{max} - 2 \times \beta_{current}
above and then rounded to the nearest multiple of 1/4096. This factor is applied if the battle occurs in dark grass, and depends on the number of Pokémon that have been caught in the Pokédex, as shown in the table below, usually making it harder to catch a Pokémon.
Number Caught	Multiplier
> 600 	1
451-600 	3686/4096 (90%)
301-450 	3277/4096 (80%)
151-300 	2867/4096 (70%)
30-150 	2048/4096 (50%)
< 30 	1229/4096 (30%)
If the modified catch rate is greater than 255, the Pokémon is guaranteed to be caught (but a critical capture check will be performed anyway).
===Shake Probability===
The formula for phi is
\varphi = \frac {65536}{\sqrt{(\sqrt{(\frac {255}{\Omega})}})}
where the divisions and square roots are rounded to the nearest 1/4096, and the final value is rounded down to the nearest whole number.
===Shake Checks===
Generation V's shake checks work essentially the same as those of Generations III and IV.
Before any shake checks are performed, the game checks to see if a critical capture may happen. If not, three shake checks will be performed, otherwise, only one shake check will be performed.
In a normal capture, the Pokémon breaks free without shaking if the first check fails or after one shake if the second check fails. If the third check fails, the Pokémon breaks free after three shakes (not two). In a critical capture, the ball will always shake once, and the Pokémon will break free or be caught depending on the result of the shake check.
==Capture Method (Generation VI)==
In Generation VI games, on the first route where a player can catch wild Pokémon (Kalos Route 2 in Pokémon X and Y or Hoenn Route 101 in Pokémon Omega Ruby and Alpha Sapphire), wild Pokémon will never break out of a Poké Ball thrown at them.
===Modified Catch Rate===
The modified catch rate is essentially identical in all respects to Generation V, except that Capture O-Powers give a 1.5×, 2× or 2.5× boost rather than 1.1×, 1.2× and 1.3×.
===Shake Probability===
Unlike Generation V, the shake probability in Generation VI is calculated as follows:
\varphi = \frac {65536}{(\frac {255}{\Omega})^{0.1875}}
===Shake checks===
The Generation VI games perform four shake checks again, like Generations III-IV. Critical captures function like in Generation V, with only one shake check being performed for a critical capture.
The change to the formula for the shake probability is designed to counteract the change in the number of shake checks: the odds of success for a normal capture are the same in Generation VI as in Generation V, barring small rounding errors. Since each shake check is more likely to succeed, however, critical captures are more effective than before.
===Critical capture===
Since Generation V, when a Poké Ball is thrown, a special type of capture that is much more likely to catch a Pokémon, called a critical capture (Japanese: 捕獲クリティカル critical capture), may occur. When a critical capture occurs, the Poké Ball will make a high-pitched whistling sound as it is thrown, then pause in mid-air, and shudder before it drops down to the ground, shaking only once on the ground before the Pokémon escapes or is caught.
The probability of a critical capture occurring depends of the number of Pokémon that have been caught in the Pokédex. First, compute the value of a (the modified catch rate) as shown in the appropriate section above. Then, multiply a by the appropriate multiplier from the table below.
Species Caught	Alpha
>600 	2.5
451-600 	2
301-450 	1.5
151-300 	1
31-150 	0.5
<=30 	0
Finally, divide the result by 6 and round down to the nearest integer. Call this final result Gamma. A random number between 0 and 255 is generated; if it is less than Gamma, a critical capture will occur. Overall, this means the critical capture chance ranges from 0 (if 30 or fewer species have been caught) to about 41.67% of the modified capture rate Omega (if over 600 species have been caught).
Critical captures only make one shake check instead of three or four, which means they are considerably more likely to successfully capture the Pokémon. In Generation V, the success chance for a critical capture is the cube root of the success chance for a regular capture; thus, for example, if the regular success chance is 5%, the chance of success with a critical capture will be about 37%, while if the regular success chance is 50%, the chance of success with a critical capture will be just over 79%. In Generation VI, meanwhile, the success chance for a critical capture is instead the fourth root of the success chance for a regular capture: if the regular success chance is 5%, the critical success chance will be around 47%, while if the regular success chance is 50%, the critical success chance will be around 84%. Given the rarity of critical captures, however, the Pokémon is always more likely overall to be caught with a regular capture than a critical one.
Formula:
\Gamma = \Omega_{CR} \times \alpha
Category:Game Mechanics