Calculate price of item using chance to receive item and original price

Question

Didn't know how else to word the title, I apologize. Basically, if I were to buy a pack of cards for 500(theoretical units), and it contains 3 cards, that's 167 per card. However, is there a way to find the average value of a certain type of card (Epic, Rare, Common) based on the chance that they will drop? Drop rates being: Epic: 2%, Rare: 18%, Common: 80%.

Answer 1

tldr: recommend commons worth $69$, rares worth $308$, and epics worth $2777$.

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There are a large number of factors here, most importantly, how much more value you actually place on each rarity. Some players have complete playsets of all commons and rares and purchase packs only for the chance of getting epics, in which case each epic would be worth upwards of $8333$ units (with commons and rares worth $0$ units each), whereas if you place the same value on each, then every card would be worth as you say $167$ units. A scale somewhere inbetween might be preferred.

Let $c$ be the value of a common card, let $r$ be the value of a rare card, and let $e$ be the value of an epic card. We have the following equation:

$$3(2e+18r+80c) = 6e + 54r + 240c = 50000$$

which comes from the concept that the amount spent should be worth the value received on average, and that purchasing 100 packs should yield 6 epics, 54 rares, and 240 commons on average and costs exactly 50000 units.

Couple this with some sort of relation between the values of each rarity, such as $e = 9 r$ and $e=40c$ in order to get that previous equation into one variable.

(This is not the only choice of relations, as mentioned, you might choose to put much less value on common cards, in which case something like $e=120c$ or more might be preferred or even putting a specific value such as $c=0$ if you'll trash them or $c=1$ if thats what you can vendor the cards for. I picked these relations since you are likely to get 40 commons in the same time you get one epic, and you are likely to get 9 rares in the same time you get one epic).

Solving using our relations, we get:

$50000=6e+54r+240c=720c\Rightarrow c\approx 69$

$50000=18e\Rightarrow e\approx 2777$

$50000=162r\Rightarrow r\approx 308$

Doublechecking, we should have $3(.02 e+ .18 r + .8 c)=500$. Indeed, it matches (albeit with small rounding issues due to using $\approx$).