I'm working on this for a while. I thought the encounter chance of Pokemons worth a bit of scientific research 
This short study tries to answer 2 questions. What is the Real Encounter Chance of a Pokemon and why is it possible to encounter with a Pokemon with the same rarity in area A than area B. I will present the topic with the case of Pikachu, because it is catchable only in 3 regions, and it has Very Rare rarity in both the Viridian Forrest and the Power Plant. Still, how is it possible that it is more rare in the Viridian Forrest? It is well-known that it's easier to catch one in the Power Plant. Why?
IN SHORT:
The more Pokemon is catchable in the area, the less chance you'll get to encounter a rarer Pokemon, so even if the Pokemon has the same rarity in two different areas, you may encounter it more in one than another.
PROOF:
When you encounter a Pokemon, the game makes two dice rolls (this is the first presumption, see the end of the article!): The first is based on the encounter chance or rarity of the Pokemon and determines which Pokemon you will encounter, the second (series of) dice roll(s) will determine its IVs and if it's good enough, the Pokemon will be shiny. We know that the so called encounter chance (EC) of a Very Rare (VR) Pokemon is 1/100. For a Horribly Rare (HR) it's 1/1000, for a Rare (RA) it's 1/10. But what would be the rate of a Common (CO) Pokemon? Logically it would be 1/1. But a popular theory says that EC means the chance of encounter in percentage. So you have 1% chance for a VR, 10% for a RA, but 1/1 would mean 100% of a CO, so you should encounter that Pokemon every time. So let's say the unknown CO rate is 1/2.
But there is some problem with the theory above. In the Viridian Forrest there are 5 CO Pokemons (Caterpie, Weedle, Rattata, Ledyba, Spinarak), 4 RAs (Kakuna, Metapod, Beedrill, Buterfree), 2 VRs (Pikachu, Pidgeotto), and 2 HRs (Schyter, Pinsir). If we ignore the uncertainity of the CO pokemons we'll see that in 52.2% we should encounter a not CO Pokemon. And 10% of that would be a Butterfree so players wouldn't complain about how hard it is to get it. And every 20th should be a Pikachu! Not to mention that following that logic, the remaining RA Pokemons would share only 47.8% chance of encounter which would mean that they have only 9.56% chance each, so they would be in fact rarer than RA Pokemons!
The problem is the misinterpretation of the rarity. It is not a percentage chance of encounter, but a proportion of the 100% chance of encounter. Let's imagine that the encounter chance is a great spinning wheel, like in the TV shows. You have different chance to spin different Pokemons, because some Pokemons have more slots than others. The wheel is always full which means that you have 100% chance to spin SOME Pokemon. There is no chance that you spin the wheel and you woN't encounter any Pokemons. But what are your real chances to see a certain Pokemon? This is determined by the rarity. Let's presume that CO rarity is 1/1. We know that when an encounter happens, you WILL meet a Pokemon, so you have 100% chance to see some kind of Pokemon. The rarity in fact is the proportionality of the chance of encountering certain Pokemons. RA is 1/10 VR is 1/100 which means that you have 10 times more chance to encounter a RA Pokemon than encounter a VR, and 100 times more than encounter a HR. That's what I call relative rarity And into this system the value 1/1 of the rarity of a CO Pokemon first perfectly.
Now let's do some math. CO's rarity is C, RA's rarity is R, VR's rarity is V and HR's rarity is H. Because we know that rarity is a proportionality we can say that
C = 10R (because you have 10 times more chance to spin a CO than a RA)
R = 10V
V = 10H
Which means:
C = 1000H
R = 100H
V = 10H
H = H
So you have 1000x more chance to encounter a CO than a HR Pokemon.
And here comes the relative rarity in. Every Pokemon will get slots on our spinning wheel, determined by its relative rarity. So on the Viridian Forrest spinning wheel a Caterpie will get 1000 slots, a Beedrill gets 100 slots, a Pikachu gets 10 slots and a Pinsir gets 1 slot. If we hand out all the slots, so every Pokemon will get slots according to its rarity we will get the real encounter chance (REC) of a Pokemon.
So in the Viridian Forrest Caterpie, Weedle, Ledyba and Rattata get 1000 slots each, Kakuna, Metapod, Beedrill, Buterfree get 100 each, Pikachu and Pidgeotto get 10-10 and Schyter and Pinsirgot only 1 each. We distributed alltogether 1000x5+100x4+10x2+1x2 = 5422 slots. 5422 slots mean 100%.
Which means that the REC of a Pokemon is:
HRs (each): ~0.0185% (0.0184337882700111)
VRs (including Pikachu): ~0.185%
RAs: ~1.85%
COs: ~18.5%
All the numbers are rounded, but if you sum it up, you'll get 100%.
From the data you can see that the Real Encounter Chance of a Pikachu in the Viridian Forrest is ~0.185%. Far more less than in the theory above.
Now let's see the RECs in the Power Plant, the other place where Pikachu can be encountered.
Here we have 4 COs (Voltorb, Grimer, Geodude, Sandshrew), 2 RAs (Magnemite, Sandslash), 2 VRs (Pikachu, Electrode), 2 HRs (Elekid, Koffing). This means that we distribute altogether only 4222 slots on the Power Plant spinning wheel.
From this the REC of the Pokemons:
HRs (each): ~0.0236%
VRs (including Pikachu): ~0.236%
RAs: ~2.36%
COs: ~23.6%
So Pikachu's REC in the Powerplant is ~0.236%. It is significantly higher than in the Viridian Forrest!
If you check the Real Encounter Chances of the Pokemons in every region, you will get the REC of all of them, and so the best place to hunt them and their real rarity as well.
Possible criticism and rebuttal of the criticism:
The whole article is based on the presumption that there is one dice roll to chose the species of the Pokemon we will encounter. But what if the game first choses from which "pool" (CO, RA, VR or HR) will it choose, and then it choses the Pokemon. We can keep the presumptions that CO rarity is 1/1 and that rarity is proportionality. In this case in the Viridian Forrest the relative rarity of the Pokemons:
C = 1000H; R = 100H; V = 10H; H = H still stands so from 1111 rolls we'll 1000 times a CO, 100 times a RA, 10 times a VR and 1 times a HR.
Consequently the encounter rate of a single Pokemon from every group in the Forrest is:
HR: 0.5
VR: 5
R: 25
C: 200
The problem with this criticism is that both the Viridian Forrest and the Power Plant have only 2 VRs. Pikachu, and Pidgeotto/Electrode. According to the criticism the chance of getting into the VR "pool" is always the same (1/100, or 0.9%) , which would mean that the chance of getting Pikachu should be the same in both regions (0,45%), because once we got into the VR "Pool", we have the same chance of getting Pikachu from it.
I hope there will be someone who reads it
This short study tries to answer 2 questions. What is the Real Encounter Chance of a Pokemon and why is it possible to encounter with a Pokemon with the same rarity in area A than area B. I will present the topic with the case of Pikachu, because it is catchable only in 3 regions, and it has Very Rare rarity in both the Viridian Forrest and the Power Plant. Still, how is it possible that it is more rare in the Viridian Forrest? It is well-known that it's easier to catch one in the Power Plant. Why?
IN SHORT:
The more Pokemon is catchable in the area, the less chance you'll get to encounter a rarer Pokemon, so even if the Pokemon has the same rarity in two different areas, you may encounter it more in one than another.
PROOF:
When you encounter a Pokemon, the game makes two dice rolls (this is the first presumption, see the end of the article!): The first is based on the encounter chance or rarity of the Pokemon and determines which Pokemon you will encounter, the second (series of) dice roll(s) will determine its IVs and if it's good enough, the Pokemon will be shiny. We know that the so called encounter chance (EC) of a Very Rare (VR) Pokemon is 1/100. For a Horribly Rare (HR) it's 1/1000, for a Rare (RA) it's 1/10. But what would be the rate of a Common (CO) Pokemon? Logically it would be 1/1. But a popular theory says that EC means the chance of encounter in percentage. So you have 1% chance for a VR, 10% for a RA, but 1/1 would mean 100% of a CO, so you should encounter that Pokemon every time. So let's say the unknown CO rate is 1/2.
But there is some problem with the theory above. In the Viridian Forrest there are 5 CO Pokemons (Caterpie, Weedle, Rattata, Ledyba, Spinarak), 4 RAs (Kakuna, Metapod, Beedrill, Buterfree), 2 VRs (Pikachu, Pidgeotto), and 2 HRs (Schyter, Pinsir). If we ignore the uncertainity of the CO pokemons we'll see that in 52.2% we should encounter a not CO Pokemon. And 10% of that would be a Butterfree so players wouldn't complain about how hard it is to get it. And every 20th should be a Pikachu! Not to mention that following that logic, the remaining RA Pokemons would share only 47.8% chance of encounter which would mean that they have only 9.56% chance each, so they would be in fact rarer than RA Pokemons!
The problem is the misinterpretation of the rarity. It is not a percentage chance of encounter, but a proportion of the 100% chance of encounter. Let's imagine that the encounter chance is a great spinning wheel, like in the TV shows. You have different chance to spin different Pokemons, because some Pokemons have more slots than others. The wheel is always full which means that you have 100% chance to spin SOME Pokemon. There is no chance that you spin the wheel and you woN't encounter any Pokemons. But what are your real chances to see a certain Pokemon? This is determined by the rarity. Let's presume that CO rarity is 1/1. We know that when an encounter happens, you WILL meet a Pokemon, so you have 100% chance to see some kind of Pokemon. The rarity in fact is the proportionality of the chance of encountering certain Pokemons. RA is 1/10 VR is 1/100 which means that you have 10 times more chance to encounter a RA Pokemon than encounter a VR, and 100 times more than encounter a HR. That's what I call relative rarity And into this system the value 1/1 of the rarity of a CO Pokemon first perfectly.
Now let's do some math. CO's rarity is C, RA's rarity is R, VR's rarity is V and HR's rarity is H. Because we know that rarity is a proportionality we can say that
C = 10R (because you have 10 times more chance to spin a CO than a RA)
R = 10V
V = 10H
Which means:
C = 1000H
R = 100H
V = 10H
H = H
So you have 1000x more chance to encounter a CO than a HR Pokemon.
And here comes the relative rarity in. Every Pokemon will get slots on our spinning wheel, determined by its relative rarity. So on the Viridian Forrest spinning wheel a Caterpie will get 1000 slots, a Beedrill gets 100 slots, a Pikachu gets 10 slots and a Pinsir gets 1 slot. If we hand out all the slots, so every Pokemon will get slots according to its rarity we will get the real encounter chance (REC) of a Pokemon.
So in the Viridian Forrest Caterpie, Weedle, Ledyba and Rattata get 1000 slots each, Kakuna, Metapod, Beedrill, Buterfree get 100 each, Pikachu and Pidgeotto get 10-10 and Schyter and Pinsirgot only 1 each. We distributed alltogether 1000x5+100x4+10x2+1x2 = 5422 slots. 5422 slots mean 100%.
Which means that the REC of a Pokemon is:
HRs (each): ~0.0185% (0.0184337882700111)
VRs (including Pikachu): ~0.185%
RAs: ~1.85%
COs: ~18.5%
All the numbers are rounded, but if you sum it up, you'll get 100%.
From the data you can see that the Real Encounter Chance of a Pikachu in the Viridian Forrest is ~0.185%. Far more less than in the theory above.
Now let's see the RECs in the Power Plant, the other place where Pikachu can be encountered.
Here we have 4 COs (Voltorb, Grimer, Geodude, Sandshrew), 2 RAs (Magnemite, Sandslash), 2 VRs (Pikachu, Electrode), 2 HRs (Elekid, Koffing). This means that we distribute altogether only 4222 slots on the Power Plant spinning wheel.
From this the REC of the Pokemons:
HRs (each): ~0.0236%
VRs (including Pikachu): ~0.236%
RAs: ~2.36%
COs: ~23.6%
So Pikachu's REC in the Powerplant is ~0.236%. It is significantly higher than in the Viridian Forrest!
If you check the Real Encounter Chances of the Pokemons in every region, you will get the REC of all of them, and so the best place to hunt them and their real rarity as well.
Possible criticism and rebuttal of the criticism:
The whole article is based on the presumption that there is one dice roll to chose the species of the Pokemon we will encounter. But what if the game first choses from which "pool" (CO, RA, VR or HR) will it choose, and then it choses the Pokemon. We can keep the presumptions that CO rarity is 1/1 and that rarity is proportionality. In this case in the Viridian Forrest the relative rarity of the Pokemons:
C = 1000H; R = 100H; V = 10H; H = H still stands so from 1111 rolls we'll 1000 times a CO, 100 times a RA, 10 times a VR and 1 times a HR.
Consequently the encounter rate of a single Pokemon from every group in the Forrest is:
HR: 0.5
VR: 5
R: 25
C: 200
The problem with this criticism is that both the Viridian Forrest and the Power Plant have only 2 VRs. Pikachu, and Pidgeotto/Electrode. According to the criticism the chance of getting into the VR "pool" is always the same (1/100, or 0.9%) , which would mean that the chance of getting Pikachu should be the same in both regions (0,45%), because once we got into the VR "Pool", we have the same chance of getting Pikachu from it.
I hope there will be someone who reads it
