My Game of Pokemon GO - Stage IV.2: Hatching

Although FastPokeMap attempted a few short-lasting recoveries, soon after developers decoded the hash key, Nantic patched and used a new key.  FastPokeMap was deemed to be a loser in this cat-and-mouse game and Twitter accounts were never restored.  With catching Lapras and Porygon in the wild becoming hopeless, the only route remaining was egg hatching.

Porygon could be hatched from 5km eggs and Lapras from 10km eggs.  Despite different statistics exist, none was collected at large enough sample size, it was the best to assume all Pokestops provided equal chances of 5km and 10km eggs and all Pokemon types were hatched with equal likelihood.  There were about 41 species in 5km eggs and 15 in 10km eggs.  The number of hatches requires for a 50% chance of obtaining a Porygon and a Lapras could be calculated using the simple equations below:

Porygon: $1-\left(\frac{40}{41}\right)^n=0.5$,

Lapras: $1-\left(\frac{14}{15}\right)^n=0.5$,

where $n$ is the number of hatches required.

The probability of hatching at least one target Pokemon versus number of hatches for the corresponding egg types are charted below.  For a Porygon (orange 5km curve), we need 28 5km hatches to archive 50% and 65 5km hatches to archive 80% chance.  Lapras (purple 10km curve) requires 10 10km eggs for 50% and 23 10km eggs for 80%.  Not bad.  However, when we get an egg from a Pokestop, ~80% chance it is 5km, but 10km eggs are rare.  The most reliable statistics probably comes from a recent Pokemon fan KSBS in Singapore [1], who kept a record for 3175 egg hatches!  He got 236 10km eggs, giving a probability estimation of 7.4%.  So 80% chance for Lapras requires 23/7.4% = 311 eggs.

In hindsight, it is most likely a wrong assumption that all 10km species are equally likely.  KSBS' data suggests Lapras has a much lower chance: 8/236 = 3$\pm$1%, compared to an average rate of 1/15=6.7%!  So the more realistic Lapras hatch curve is the blue curve; one needs 47 10km eggs for one Lapras, which translates into 47/7.4% = 635 egg hatches!  This estimate matches my own experience and what I read recently on Silph Road.

Either 311 or 635, with an average 5km-walk per egg hatch, you would have to walk 1500-4500km for a Lapras.  It was pretty obvious to me that this was time to pour cash into the game -- one had to buy incubators.  The game came with only one infinite-use incubator, I needed to buy eight 3-use incubators at $1.50 each, then I could reduce my walk by nine fold using parallel processing.  Cash was injected, no sight of Porygon and Lapras.  I passed the 50% hatch point, then the 75% line, still trying ...  Eventually I got two Porygons, both with IV above 85%.  

By Dec 16, I have walked over 500km and hatched 346 eggs and I decided to quit.  With $0.50/hatch, you can figure out how much I spent in this hatching stage without accomplishing the goal.  One common mistake people tend to make, including myself, is the intuition that our past hatch failure increases the chance of future success, i.e., the more failed hatches, the next one is closer to be Lapras.  However, textbook tells us a different story, after each failure, I still need another 311 or 635 eggs in order to have 80% chance of gaining Lapras in my next attempt.  The history of 346 eggs has no effect on the future and all those incubators were simply a waste without leaving a footprint.  Getting Lapras this way is a lottery most of us cannot win.

Chasing Snorlax

Catching a rare without trackers was nearly mission impossible, but I did manage to catch a Snorlax one night 100% on my own effort.  As I walked entering the "Snorlax Detected" point from the north, the big fat fellow appeared in my sightings, which means it was 120m away.  I started running zigzag as the green arrows indicated on the map.  The yellow dash line outlined the approximate 120m boundary where Snorlax disappeared from my sightings, which made me think either Snorlax has despawned or I have moved too far, fortunately it was always the latter.  After an intense 10-min blanket scanning, a colossal Snorlax, weighted 693.05kg, entered my field of Poke vision!  Postmortem analysis certainly suggested a much better alternative tracking strategy exist, but I was not going to repeat that again, it was simply too tiresome.

Snorlax caught by brutal-force blanket scanning on Nov 11, 2016.

Reference

1. https://www.reddit.com/r/TheSilphRoad/comments/5rsyqn/coming_out_lvl_38_3175_eggs_hatched/