The odds of getting that "ideal" Pokemon that you have always wanted - 0.16%
Example:
What are the odds of getting that "perfect" Pokemon that you have always wanted via hatching (one egg)? Let's pretend that you wanted a high-IVs Kabutops with Mud Shot and Stone Edge, and that you WILL get an egg from this PokeStop ahead of you...
1/3 (see note A) chance of that egg being a 10KM egg
1/17 chance of Kabuto hatching from that 10 KM egg
1/2 (see note B) of Kabuto having 82% or greater IVs
1/2 chance of Kabutops knowing Mud Shot
1/3 chance of Kabutops knowing Stone Edge
Multiply all the numbers above:
1/612 or 0.16% is your chance of getting the "ideal" Kabutops that you wanted, after your hard work of walking 10KM to hatch the egg and gathering enough candies to evolve the Pokemon.
The math may not be perfect. The example above is merely to show how DIFFICULT it is to get your "ideal" Pokemon. This is not to say that there truly is only a 0.16% chance of obtaining this Kabutops, because you can sway the odds in your favour by hatching MORE eggs and actually go out and CATCH some Pokemon, or maybe my math was a bit off to begin with. Let's pretend that you do everything you can, and the odds are now increased TENFOLD. 0.16% x 10 is still only 1.6%.
It takes effort and LUCK to finally come across a "perfect" Pokemon. You get disappointed when your 10KM egg hatches a Pokemon that you didn't want, or when your good IVs Pokemon evolve to get bad moves; but when you finally get EVERYTHING right (like a SLOT MACHINE WIN), it feels great and you would then heavily invest in that awesome Pokemon. This is the thrill and agony that comes with playing PoGO, some players accept it, some players hate it.
Personally, I accept and enjoy the difficult pursuit of a "perfect" Pokemon. However, when a player finally gets a "perfect" Pokemon and then one unexplained, unannounced re-balancing update from Niantic adversely affects it in any way (whether it is turning that Pokemon into complete shit or just making it slightly sub-optimal), it is very dispiriting to the player. Many, including myself, find this to be unfair UNLESS Niantic introduces some kind of flexibility in "re-customizing" and "improving" a player's best existing Pokemon.
Note A: In actuality, the odds of an egg being a 10KM should be SIGNIFICANTLY lower than 1/3. I used 1/3 for convenience purposes and it is a very very optimistic assumption.
Note B: In actuality, the odds of a hatched Kabuto having 82% or greater IVs should be considerably lower than 1/2. I conservatively used 1/2 for convenience purposes.
[EDIT: Note B is found to be inaccurate (see comments below) but this should not affect the overall calculation significantly]
Answers
Well the first thing you should know is whatever pokemon you get is determined on the pokestop spin. It may happen to be one that is hatched from a 10k egg or it may not. It's a specific pokemon you get when you spin, and it is represented by the egg it hatches from. It just seems rare because 10k pokemon are supposed to be more rare. I really hope this helps. I agree with the frustration that comes from the randomness of this game, it's unacceptable.
The math is a little different actually:
First the pokestop determines if it's conna give you an egg. Then the pokestop determines which pokemon it's gonna give you and gives you the corresponding egg, then, the chance of getting the IV's you want (Sum the odds of each combination that produces the desired %), then the odds of getting the moveset you want. then you multiply them all and you get the actual probablility
I haven't done the math, but the pokestop determining the pokemon and give you the egg based on the pokemon, should improove the odds compared to your math, as you don't have 2 separate events:
Assume 10% of getting a 10K and 5% of getting a specific pokemon out of that egg, that's 0.005, instead, you'd have a single event with say... 1% for a specific pokemon. This obviously is affected by which pokemon you want. As I don't have the probability list handy, I can't give you the exact math.
your numbers are pretty generous to be honest.
1/3 and 1/17 means every egg has a 1 in 51 chance (or about 2%) being the 10k egg pokemon that you desire, which is a bit off. plus pokemon in the same egg category might not be evenly distributed, which would further lower odds.
besides someone else correcting the mistake on iv probability, people who want the best of the best don't settle for 37/45 ivs, that would be considered low ivs by them, so even if 1/2 chance was correct for those ivs, the bar for "ideal IVs" is set higher, like 41+, so the probability would therefore be lower.
which is good, because if 0.16% is considered very generous odds, that helps with people not spreading bad advice like "wait for a better one" when people ask what they should do with their pokemon that have good but not perfect moveset and/or ivs.
I tried to be as conservative in my calculation as possible. So that even if my math was so off, say off by a factor of 50, my message would still have a valid point. 0.16% x 50 = 8%, which still isn't considered amazing odds.
I am not impressed when people tell me to "just go and catch another one" after a Niantic update changes the PoGO battle scene. That is like telling me to "just go and get another slot machine win".
