"270 km (170 miles) for a Pichu with a hat – less if you are lucky – more if you are unlucky"

#PokemonGO: TL;DR A Pichu with a hat is likely the first truly rare Pokemon which less supply as there are trainers.Edit1: I get multiple anecdotal stories of Pichu as first hatch. Actually I also had one such hatch with my 2k egg (albeit an old egg). I added two columns that assume a double chance. In this case it would be 136 km or even less. We will find out with more data.The collectors among us will try to keep their collection complete. Trading isn’t possible yet – so how much effort is it to get a Pichu with a hat? I did some calculations – so you don’t have to make them yourself.Assumptions: I start with the old hatching chance for Pichu as reported on the SilphRoad - http://ift.tt/2qG61Ir . I’m using a chance of 4/255 as I assume ultra-ultra-rares are correctly predicted in footnote 1. I then add up the egg distance for each species with the number of hatches I expect. This would be 1x5 km for a Shuckle, 8x2km for 8 Pichu and 16x5km for 16 Phanpy, etc. Summing this up I get a total of 2170 km that you need to walk to hatch each ultra-ultra-rare once, each ultra rare twice, each rare 4 times, each uncommon 8 times and each common 16 times.You don’t need 8 pichu with a hat – so 2170 / 8 = 271.25km is enough. This is the average distance that you need to walk to get a Pichu with a hat. Off course this is average distance to walk. We can assume an average distance of 4.25km for each egg. This can be translated into a table that lists our chance to get a pichy versus the distance we need to walk:Chanceeggsdistanceeggs double chancedistance double chance10%73041720%156483430%2398125140%33140177250%44187229460%582472912370%773273816280%1024345121790%1466217331095%1908089540498%248105412452799%292124114662199.5%3361428168714The table gives the chance to get a Pichu with a hat, the number of eggs you need to hatch to exceed this chance and the estimated km you need to walk. You can see from the table that you need to walk an average 1054km for a 98% chance - or to turn it around - even if everyone walks >1000km there will be still one out of 50 trainers who won't get a Pichu with a hat.What can you do to increase the chances? Incubators of course suddenly look much more interesting again. But this can be pretty costly. Something that does help - keep all your 5km and 10km eggs until the end of the event and only walk 2km eggs (unless you have none left). This will gain you 3km less walked for each 5km stored and 8km for each 10km not walked. So if you get 10km eggs and don't want to use/buy incubators then you can shave up to 56km (8 10km stored) off the total to walk by postponing the distance after the event.I would like to add that all the numbers might be wrong by a factor of 2 (or more) if Niantic increased the chance of a Pichu during the event. It is well possible that this happened – but we likely won’t be able to tell until half way into the event.Oh – and one more note: If you are lucky and get 2 Pichu (or more) with a hat – do not evolve them. Wait to catch some Pikachu’s with a hat – there are likely enough around.Just a final note: I know how passionate all players here are. Please keep the discussion polite even if you are disappointed that you might not get a Pichu with a hat prior to trading.Only for the Mathematical inclined. I use the binominal distribution with a chance of 4/255 to calcultae the chance of 1 or more Pichu with a hat. I did this in Excel using =1-BINOM.DIST(0,N,4/255,TRUE) with N being the first number that is above the % reported in the table. The chance of 4/255 is likely wrong as we had a change in egg distribution. In the past the total chance stayed approx. in the same ballpark unless a species moved to a new rarity class in which case the value could be easily wrong by a factor of 2. via /r/TheSilphRoad http://ift.tt/2uyZ8Hj