How many packs to get all common and rare MSoG cards

hearthstone

How many packs of Mean Streets of Gadgetzan cards would I need to open, on average, to get 2x of all common cards and 1x of all rare cards? Additionally, how many different epics and legendaries would I get from those packs?

Best Answer

You are likely to hit the 2x of each common before you meet the 1x of each rare requirement. I won't bog you down with detailed mathematical formulae, but if you're interested, you can read more here: https://en.wikipedia.org/wiki/Coupon_collector%27s_problem

http://hearthstone.gamepedia.com/Card_pack_statistics cites a card being common as approximately a 71% chance. There are 49 unique commons in MSOG. If you were to open 60 packs, you would be expected to open [0.71*(60*5)] = 213 commons total. Looking at the coupon collector chart, the number of commons we need, on average, to have 49 unique commons is 220. We're not there yet, but we're close.

Rares, however, are substantially rarer, with most studies showing around a 23% chance for a given card. This means, with 60 packs, you're expected to open 69 total rares. There are, however, 36 rares in MSoG. So you're opening just over the amount of commons you need if you could just select the ones you wanted from a pool once you turned over a "common" rarity, but you're not even opening the amount of rares you need if you could select from a pool. Looking at the chart, the number of tries we need to have a complete set of 72 rares is 151. We're about 40% of the way there.

So, how many packs do we need to open to get a complete set? Solving for x, where 220 is the absolute maximum number of cards we need for two of each common, we can say [0.71*(x*5)] = 220, which after some algebra equates to 62 packs needed for 2 of each common. For rares, it's [0.23*(x*5)] = 151, which equates to 132 packs.

Thus, you need 132 packs to, on average, open one of each rare and two of each common through sheer discovery alone. But you'll have a ton of extra commons rares if you do this. That's a ton of dust worth of extra cards. As rares are 100 dust each and commons are 40, it's obviously substantially more cost effective to use dust to craft the last few cards.

EDIT: I made a obvious math error by assuming to collect a playset of 2x cards would be double the under needed to collect 1x. That's obviously not the case, as you would have collected duplicates of many cards before you finally found the last card you need, so only in the very worst case scenario would it be double.