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Pokemon is a video game series managed by The Pokemon Company, a subsidy of the video game company Nintendo. Over the years, Pokemon has become known to almost every person in the world. Initially, Pokemon was a popular and successful video game franchise, but, as it grew more successful it was adapted to a tv series, a toy company, and many more. As a child, I mainly spent all my time watching TV or reading comic books. One of my favorite TV shows was Pokemon. Whenever it came on TV, no matter what time, I would go to watch it. I remember loving it so much to the extent of making up my own Pokemon. Pokemon began from a wide series of best selling video games. In those games, you were faced with challenges of capturing different creatures called Pokémon.For every Pokemon that a player catches, that Pokemon’s data is recorded into a Pokedex. If a player catches all of the Pokemon in the game, then they would receive a virtual certificate congratulating them on catching them all. The problem with this is that, as of right now, there are 807 Pokemon to catch.Every person who plays Pokémon yearns to catch all the Pokemon, but, the process of catching pokemon is very tedious. I too have attempted to catch all 807 pokemon, but failed in all of my attempts. There are many procedures one can do in order to increase their chances of encountering a catching a Pokemon, however, many of them are tedious and convoluted. Are the chances of catching all Pokemon unattainable? In this investigation, I will be exploring the mathematical relationship between the probability of encountering and capturing a set of Pokémon in Pokemon Emerald. I will be focusing my efforts on encountering and successfully capturing a favorite Pokemon of mine, the Pokémon Tentacool. In order to collect data for this investigation, I will apply data into the game ‘s various formulas used to determine probabilities.

In order to create a conclusion to this investigation, I will first find the chances of encountering a specific set of pokemon, the probability of encountering that pokemon, and my chances of capturing the encountered pokemon. I must make sure that this investigation takes place in the same area at all times in order to keep variables such as encounter rate and catch rate the same throughout this investigation.

In many areas across the game, select Pokemon have percentages of encountering it. This percent value is the probability of having that Pokemon encounter with you. The Pokemon set, I have chosen to find and capture consists of 3 pokemon. The Pokemon, Wingull has a encounter rate of 7/20, Tentacool has a rate of 3/5, and Pelipper has a rate of 1/20.

The catch rate of a Pokemon determines the players chance of catching the Pokemon encountered. The rate is out of 255. The formula shown below is used to find the catch rate of catching a Pokemon with any possible factors that may arise in a battle.

a = catch rate

HPmax = The maximum amount of hit points for that Pokemon (I will be using either 1 or the max health for each Pokemon.

HPcurrent = The current amount of hit points for that Pokemon

Rate = The catch rate for that Pokemon

Bonusball = Multiplier for that ball used to capture the Pokemon

Bonusstatus = Multiplier for any status ailments that the Pokemon may have

The location of our investigation determines the encounter rate of every Pokemon available. In some areas of the game, the encounter rate for each of these pokemon drastically fluctuates. For that reason, I will be focusing on searching throughout Route 105.

The probabilities for each of the Pokemon I will be searching for are within the table below In the table, the first row lists the lists the encounter probability. These values are preset and remain the same throughout this investigation. The remaining rows lists the catch rate probability, depending on the ball used to catch the Pokemon, and the third is the catch rate affected with any status ailments. The catch rate value varies under certain circumstances. According to the catch rate formula, the variables Bonusball and Bonusstatus affect the catch rate value. Depending on the type of ball used to capture the Pokemon, the catch rate will either decrease or increase. The Bonusstatus variable is only affected if the Pokemon has a status ailment such as burn, frozen, paralysis, poison, and sleep. Below I have listed the numerical effects that these variables would have. To find the catch rate for all possible events, I substituted all pertinent values to the provided catch rate formula.Also the probabilities of each pokemon if they were to be encountered at their maximum health.When encountered, the catch rate value is different from Probability values of If the pokemon were to be at 1 health, are listed in the table below.

Based on the values listed within the tables earlier, the probability of encountering and catching either one of the Pokemon in almost every scenario, is able to be determined. Through the use of probability theory, it is possible for one to use the associated formulas and find the probability of anything. For this investigation, the two relevant formulas needed are the P(A∩B) and P(B|A). The first formula is used for events, whose occurrence of either of them doesn’t affect the probability that the others occur. For this investigation, I can find the probability of encountering and catching the pokemon being sought after. Below is an image that further explains the P(A∩B) formula.

P(A∩B) = P(A) ⋅ P(B)

In my case, P(A) can be used to represent the encounter rate of a Pokemon. P(B) can be used to represent the value of the catch rate. The catch rate value will be the values listed in the previous tables, that way I can find all the probabilities of all possible outcomes, thus determining their likeness of success. The other formula would be used for more specific events. In the formula, P(B|A), I would be able to find the probability of outcome success if I were to use different factors, under different circumstances. The formula for P(B|A) is listed below. In this equation, P(B|A) represents the probability of B occurring given that A has already taken place. Where B is the event, and A is all the events that had taken place before B to happen. The table below presents all the values for P(A∩B). The values for poisoned, paralyzed, or burned status ailments are differentiated by the asleep/frozen column next to them. Therefore, the first column to the right of Asleep/Frozen Max health, is meant for poisoned, paralyzed, or burned status ailments at max health. In order to attain these values, I simply substituted the respective encounter and catch probabilities for each Pokemon, as well as the catch rate if there were any factors that would have affected this value such as status or health amount. Below the general process that had taken place in order to determine the value for every probability. Based off the data found using the P(A∩B) formula, it can be said that for the best chance at catching a Pokemon, they must have 1 health and either frozen or asleep. For the best possible outcome, one must use a Dive ball. However, Dive balls are quite expensive to buy and scarce to find out side of stores. Due to this, the use of other balls is needed. Through the use of the P(B|A), one is able to find the success rate of using other balls to use given that the Pokemon is at 1 health and either frozen or asleep.Below is the general process that would’ve taken place to determine the values. In this example,I first found the probability of even catching a Pokemon with a Pokeball. Finding this value was quite simple because, all I had to do was add all the chances to catch a Pokemon with a Pokeball, which are listed in the previous table. After, I divided the value of catching a Pelipper at 1 health and while it is frozen or asleep, with the value of catching a Pokemon with a Pokeball.

In the table below are listed the probabilities of conditional events, each labelled by the conditional statements as their respective table titles.In each table, there is a blank space that is meant for the type of Pokemon that is being used.Within the columns that displays the probability, I have listed the overall probability of each ball. All ball capture values are listed alongside the respective pokemon.

From this investigation, I was able to determine the best possible factors to ensure my success in capturing a Pokemon. However, to determine a more specific probability for catching Pokemon, one would have to look at more random variables. In the game, there are many random events that aren’t able to be determined such as gender and IV’s(Individual Values). Gender affects a pokemon health , which is a key variable in the catch rate formula, however the extent to which the health differs between two pokemon are ultimately defined by a Pokemon’s IV’s and EV’s. IV’s are instrumental in determining the stats of a Pokemon, being responsible for the large variation in stats among untrained Pokémon of the same species.There is no way to predict these values, there are too many factors to consider when attempting to make a truly accurate prediction. Despite this fact, this investigation has made me realize just how math such as probability can be applied to something such as video games. By furthering my understanding of the topic, I hope to determine more accurate probabilities for this game.

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