Welcome back, readers! Recently I noticed a post in the forum by undercoverlucky requesting a more in-depth look at EV (expected value). Before we delve too deeply I wanted to make a special thank you to one of our fellow QSer's, WeQu, whose help was greatly appreciated as I'm no data miner at heart.
Here's one definition of our topic at hand:
"Expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity."
In relation to cracking Magic packs, this consists of determining the average value of each pack (typically with emphasis on the rares).
Calculating the EV of a Pack
Step 1: Determine the estimated value of all rares in the set.
To accomplish this we'll need to add all the current values and then divide by the total number of rares. We'll want to do the same for mythics, though our calculations will need to include the fact that mythics are less likely to be opened. One could in theory continue this trend to include uncommons (multiplying the estimated value by 3, although that wouldn't take print runs into account) and similarly commons. However, given that the estimated value of commons and uncommons for a set is incredibly low, we typically ignore these calculations. Foils are a different challenge all together.
Here are the current estimated values at TCG mid (on 5/9/13) for the current Standard-legal sets:
|Set||Mythic Rare||Rare||Uncommon||Double-faced Mythic||Double-faced Rare|
|Return to Ravnica||$5.50||$2.10||$0.23||-||-|
Step 2: Find the average aggregate value of the rare slot
This is accomplished by taking a weighted average between the mythics and the rares. The equation is:
Weighted Average = x1 * w1 + x2 * w2
x1 and x2 are the mythic and rare average values, respectively, and w1 and w2 are their proportionate weight. The probability of pulling a mythic in most booster packs is about 1/8, so that makes the weight for mythics 1/8 (0.125) and that for rares 7/8 (0.875).
For the current set, Dragon's Maze, the weighted average is ($10.50 * 0.125) + ($1.87 * 0.875) = $2.95
Step 3: Add the average value of the uncommons
This step is optional, as discussed above. It should yield a more accurate result, but it's important to note that average TCG values for many uncommons can be difficult to find. (This is one reason people often don't factor uncommons into EV calculations.)
Take the rare value above and add the average uncommon value ($0.29) multiplied by 3. This gives us roughly $3.82 a pack, which is right around MSRP. By this calculation, if we can purchase packs at MSRP and sell every rare and uncommon at its average TCG price, we should about break even.
It is crucial to note that this phenomenon occurs with almost every new set, but the average value will drastically decrease as more packs are opened and supply meets up with demand. If you run this calculation in about three weeks, you'll most likely find the average value of Dragon's Maze drastically lower than $4.00. This can be verified looking at an older set:
Avacyn Restored Weighted Average = ($6.56 * 0.125) + ($1.69 * 0.875) = $2.30.
After adding three times the average uncommon value ($0.22) we have roughly $2.96 per pack.
Large Sample Sizes Required
It's important to remember this concept is based on the idea of opening a very large number of packs. I mention this because most people buy just a few boxes, from as little as one or two. If you are in that camp, you want to alter your calculation for a smaller sample size.
Basically, you want to isolate and separate the outliers, the few chase rares and mythics that keep the mean average of a set up. In these cases, it's wise to chart current prices on a bell curve and eliminate the largest outliers on the mythic rare scale. You tend to get only about four per box, out of a total 10-15 mythics in a given set. This means the probability you see one of the more valuable ones is between 26.66% and 40% (depending on the number of mythics and how many are expensive.)
So now you know how to calculate the estimated value of a pack.
The funny thing is that we often estimate a rough value without even doing the equations. Typically when you look at a set's rares and see a lot of $0.5 to $1.00 cards your brain lets you know that you're more than likely not going to get your $4.00 worth out of a pack. This is why with the exception of draft and sealed you'll rarely see competitive players cracking packs. It's usually cheaper to buy the cards you actually need directly, rather than risk pulling a bunch of bulk you don't.
On a Side Note
Cracking packs is still fun (at least for me) and sometimes a necessity, like when a set first comes out. Personally, the only packs I get come from pre-ordered boxes (because the cost per pack is closer to $2.70 or $2.44, depending on how many boxes I buy) or prize packs from tournaments. Outside of (pre-)release events I don't really enjoy drafting or sealed, but that's a personal preference.
One factor unrelated to estimated value that one should consider when determining whether to buy packs/boxes is the opportunity cost for trades. The best time to trade is right when a new set comes out, as all types of players want to get their hands on the new cards. This is why I crack prize packs I receive during the first few weeks of a set's release; any other time I'd rather trade them in for store credit to pick up singles.