Hey guys! This is my first post here on SteemStem so any feedback regarding format or the mathematics in general would be greatly appreciated. Let's start exploring the new field of Steem Monsters Mathematics!!

Introduction to Steem Monsters

I assume everyone here on the blockchain has heard of Steem Monsters already, but if you haven't, feel free to log into Steem Monsters with my affiliate link, this will give me 5% commission on your future purchases! For those who haven't heard of it yet, it's basically a TCG that is powered by our very own Steem Blockchain! I recently acquired a lot of alpha packs, and was interested in the expected value of opening them all compared to selling them, so thus began my quest to find the expected value of each pack!

Calculating the value of a pack Steem Monsters pack

This seemed like a very simple task at first, just multiply the value of each card by its expected drop rate, right? Well, it's not quite that simple.

Calculating the drop rates of each rarity

First, we'd have to calculate their drop rates. We have 75.2% common, 20% rare, 4% epic and 0.8% legendary. They each have a 2% chance of becoming a gold variation of that rarity after they are revealed. That information can be summarised in the table below.

However, we must also be aware that every pack of 5 cards is guaranteed to have a card that is rare or above, but what's the chances of there being 5 commons? Actually it's quite high, 0.752^5 = ~0.2405.

This is where my mind started to get a bit confused since I initially just subtracted 24.05% from from the chance of commons and spread the probability out to the other rarities using their respective drop chance as a weight, but then I realised that is VERY wrong. Since it's a 24.05% chance every 5 cards, 1 common is turned into a rare, epic or legendary, we have to say that out of every 5 cards, there's a 24.05% one of them is turned into a different rarity. Does that mean we can just divide 24.05% by 5 and spread it out to the other rarities using their drop rates as a weight? Yes.

We know cards are opened in packs of 5, so we'll only ever open multiples of 5 cards. The above only happens when we open 5 in a row that is common (in same pack), if 4 in a row in a pack THEN the next card opened in the next pack is also common, it doesn't trigger this effect. Therefore, it is valid to simply divide 24.05% by 5 and spread out the percentages across the other rarities. Below is table with the newly constructed probabilities.

HOWEVER, THE ABOVE TABLE IS WRONG!

Just as I was about to hit the submit button, I realised that the rarities for the gold drops do not change, since in my Excel sheet, I calculated the gold drops first, then multiplied that by 49 for the normal variation. However, when I was accounting for the at least one rare feature, I added the weights to the wrong side.

The correct table

This table now tells us the true percentage of each rarity. All we have to do now is multiply it by the average value of each rarity.

• Note: We can verify this table is correct by checking the right column. We expect common drop rates to be around (75.2%-(24.05%/5)) = ~70.39%, which is consistent with our table. Since this is correct, our weightings are just constants so the rest should also be correct since it adds up to 100% on the bottom.

Calculating the value of each rarity

This part is much simpler, I could just add the price of every card from each section and divide it by the total number of cards, thus finding the average. However, I have chosen to use the median lowest price/BCX for now and an estimate for golden legend prices as a more conservative estimate.

Here's how much value each tier of cards adds to the total value!

AND FINALLY, the expected value of an alpha pack

After multiplying the price by the probability then adding it all up, we have come to a final price of $3.20 per pack! That's 4.09 Steem per pack at a price of $0.783 per Steem as a conservative estimate!

If you're interested in buying some alpha packs, check out this article here!