User blog:Tecku/Tecku Talks: Card Pack Value

So, if you don't know, today's daily challenge in PvZ: Heroes is impossible. PopCap is using the same Puzzle Party from last year but didn't take the nerfed Medula Nebula into consideration, making it literally unbeatable. Maybe PopCap will give us a bonus pack for this error. Speaking of which, what is the value of a pack, anyway?

Tecku Talks About: Pack Value

This post will use some math to determine the average spark generation rate per pack. For starters, let's calculate the average value of any given pack, ignoring elements such as getting new cards, the small chance of unlocking a Hero, and bonuses from multipacks. Each pack has a guaranteed Rare card, a 30% chance of getting a Super-Rare card, and a 10% chance of getting a Legendary card. These values are independent of each other as well, so there is a 3% chance of getting both a Super-Rare and a Legendary card.

Assuming we get all duplicates, the spark value is calculated as the chance of each card occurring times their value: This adds up to about 294 sparks in the average pack of cards, assuming all are duplicates. However! If we are saving our Gems and spending them on multipacks, this number jumps up to 323.4. That means that if we save our Gems up, every group of 100 Gems becomes worth 323.4 sparks.
 * 50- (One Rare card)
 * 45- (at least Three Uncommon cards)
 * 75- (0.3*250 chance of hitting the Super-Rare)
 * 10.5- (0.7*15 of getting an Uncommon)
 * 100 (0.1*1000 chance of hitting the Legendary)
 * 13.5- (0.9*15 of getting an Uncommon)

HOWEVER! This number increases the more incomplete our collection is. If rather than getting a duplicate Legendary, we get a new Legendary, we can assume that the card is worth 4000 points instead of 1000. This, of course, means that the value of the packs (are by extension the Gems used to buy them) become worth more.

Let's assume that we have all Super Rares, and are only getting new Legendaries. Since the odds of getting a new Legendary change over time as our collection fills, we shall assign it a variable. In this case, we will use the letter L to stand for the odds that a Legendary will be new. Most of the values in the above calculation stay the same. However, the lines for hitting the Legendary is replaced with these new lines: This means the value of 100 Gems put into a pack is 294+300L, where L is the fraction of Legendaries you have yet to max out. Add the multipack bonus, and we get 323.4+330L instead.
 * 100-100L (0.1*1000*(1-L), where L is the chance that a Legendary will be new)
 * 400L (0.1*4000*L, where L is the chance that a Legendary will be new)
 * 13.5- (0.9*15 of getting an Uncommon, stays the same)

So, let me ask you: when does it become more efficient to buy Legendaries for decks outright instead of buying packs?

First, let's calculate what the value buying Legendaries outright is. Legendaries are worth 4000 sparks and cost 1150 Gems when bought outright, so the spark value is 347.826 per group of 100 Gems. In order for it to be more efficient to buy cards outright, this value must be more than the previous bonus. Thus, 347.826 <= 323.4 + 330L. Plus that in and we get the 0.074 <= L.

If the fraction of Legendary cards you want comprises less than of 7.4% of the Legendaries in the pack, it is more efficient to buy them outright. Otherwise, your better bet is to buy multipacks.

Let's give an example: say you really, really want a Dark Matter Dragonfruit, no matter what. Dark Matter Dragonfruit comprises 5% of Galactic Legendaries, so it's better to buy it outright.

However, say you want a Dark Matter Dragonfruit OR a Supernova Gargantuar. These make 10% of all Galactic Legendaries, so it's more efficient to buy the packs.

This also means buying multipacks is more efficient than buying cards for Triassic Triumph and Colossal Fossils outright since they can only go to a minimum of 10% chance for a single Legendary.

So, yeah. Hope this is helpful!

P.S. The calculation including Super-Rares is 294 + 225s + 300L for regular packs and 323.4 + 247.5s + 330L for multipacks where s equals the percentage of desired Super-Rares and L equals the desired percentage of Legendaries.