The Secret of Egg Hatching

The mystery behind egg-hatching has been unveiled by Silph Road researchers (here) [1]!

For a long time, the understanding has been a Pokestop picks a 2km/5km/10km egg based on an egg-probability distribution, then pick a Pokemon within that distance according to another probability distribution.  For example, the chance of getting a Lapras is $p(Lapras) = p(10km)*p(Lapras|10km)$.  Now there is a much simpler and elegant new theory: eggs are simply determined by one probability distribution - one that is associated with Pokemon alone.  Hatchable Pokemons are assigned to four categories: Common, Uncommon, Rare and Ultra Rare, with hatching probability ratio 8:4:2:1.

The assignment of Pokemons based on their hatch distance and rareness is shown in the table below [2]:

Converting this chart to counts:

Rareness	Weight	2km	5km	10km	Total by Rareness (weighted), Row Sum
Common	8	6	10	2	144
Uncommon	4	16	15	3	136
Rare	2	1	4	3	16
Ultra-Rare	1	1	13	5	19
Total by Distance (weighted), Column Sum		115	161	39	315

Imagine there are a total of 315 eggs in a basket, Lapras (ultra-rare) has only one egg, Geodude (common) has eight eggs. Pokestop randomly picks an egg from this basket, distance (egg color) is just a by-product. Then the probability of hatch a particular Pokemon can be calculated based on its category (regardless of its egg distance):

common: 8/315 = 2.54%,
uncommon: 4/315 = 1.27%,
rare: 2/315 = 0.63%,
super rare: 1/315 = 0.32%.

The egg drop probability from a Pokestop can also be explained:

2km: 115/315 = 36.5%,
5km: 161/315 = 51.1%,
10km: 39/315 = 12.4%.

For Lapras, the chance of getting a 10km egg is 12.4%, the chance of that 10km egg being Lapras is: 1/39 = 2.56%.  The overall probability of hatching a Lapras is 1/315 = 0.32%.  This is quite consistent with our previous observations (see my previous Blog entry on hatching).  We can further infer that to hatch a Lapras with a 50% chance, one should hatch 218 eggs, including 27 10km eggs.  To hit an 80% chance, one should hatch 506 eggs, including 62 10km eggs!

Let us check on Gen 2 babies. Babies are all uncommon with a probability of 1.27%, despite of their distances.  So to hatch a particular baby, one needs to hatch 54 eggs (50% chance) or 126 eggs (80% chance) (I hatched Tyrogue with 108 eggs, in agreement).  Pick Tyrogue in particular, one should walk 25 or 59 5km eggs in order to hit 50% and 80% chance, respectively.  Although these many eggs would probably cost you $50 in incubator purchase, it is still way more feasible than hatching a Lapras.

This is indeed a beautiful theory that explains all the mysteries behind eggs! Congratulations to the researchers!

Reference

[1] https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&cad=rja&uact=8&ved=0ahUKEwiBroGQzczSAhXGjFQKHZckDJoQqUMIMjAH&url=https%3A%2F%2Fthesilphroad.com%2Fscience%2Fsecret-egg-rarity-tiers-pokemon-go&usg=AFQjCNHhNlqpvW0rLfDNP8byWlLfOhPA5A&sig2=6hWcKtL7eeiMm71pxKNA7w
[2] https://www.reddit.com/r/TheSilphRoad/comments/5ylpzj/chart_of_egg_rarity_divided_by_egg_distance/