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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 |
---|---|---|---|---|---|

Dragon's Maze | $10.50 | $1.87 | $0.29 | - | - |

Gatecrash | $5.00 | $1.91 | $0.27 | - | - |

Return to Ravnica | $5.50 | $2.10 | $0.23 | - | - |

Magic 2013 | $7.33 | $1.31 | $0.35 | - | - |

Avacyn Restored | $6.56 | $1.69 | $0.22 | - | - |

Dark Ascension | $4.51 | $1.18 | $0.25 | $10.84 | $0.37 |

Innistrad | $6.31 | $1.93 | $0.24 | $9 | $1.28 |

#### 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.

Nice article! Is foils too difficult to account for that it’s better to just look at it as a bonus?

I do see them as a bonus when I calculate EV 🙂

Considering the ones being worth money are much more rare compared to their non-foil versions, will only make the spread from the sample mean too far away.

Small example = 9x $5 + $100 makes $14.5 average, but only 10% actually has value above $14.5 and 90% way under it…

-> Working with averages can make you feel disappointed if you take foils into account.

FYI – I noticed the actual EV of booster packs were missing in this article, so I’ll be listing the EV values based upon today’s TCGPlayer values somewhere in the forum today (I’ll provide a link here)

Ya…I didn’t calculate the EV for each booster mainly because I wasn’t sure how people would want to cover it (i.e. include uncommons or not) so I put in the equations and gave people the numbers and figured they could calculate it based on how they wanted to. I feel that including uncommons is sort of “lying” to yourself because the avg uncommon is $0.25 to $0.3 and since you get 3 of them that’s basically an extra $0.75 to $0.9, which is almost 25% of your pack cost; yet the demand for any but a few of the uncommons is usually non-existant and you’ll have a very difficult time unloading them.

Tomorow, my article will be about EV too. I had make a part on booster, but I removed it when I learn you wrote your article on that topic.I think I can add 2 or 3 things…

First, when you say that we must not factor the commons and the uncummons (or that it’s optional) really depends on the set you are calculating your EV. My example was based on a call from Matt Lewis about Urza’s Legacy boosters on mtgo. Cloud of faeries is one of the common in this set and is worth 8$. Some other commons are 2-3$ in this set too (a lot of pauper is played on mtgo). So here, the EV of the pack is almost exclusivly driven by the price of the commons and not the rares!!!

Second, one of the biggest mistake about EV (and I’ll talk about it tomorow) is exactly what you mentionned about needing a big sample. Poeple think that because they have +EV that they will make money for sure. In the long run, if you always buy +EV packs, you will make money. Sill, if you have the chance to buy +EV packs, even if it’s a small sample, you should do it! Since by doing so you are doing money in the long run…

Finally, I had some issues with foil calculation too. you’ll have one foil every 63 or 64 cards (I don’t remember wich one) and it’s the only time you can have 2 times the same card in the pack (the foil and the regular one). I think that counting that has a bonus is a really really good idea and works well for an approximation!

good article

Thanks. I agree with the foil issue. I’m sure it’s do-able, but the probability is low enough compared to the value added that if we were to do all the calculations, I honestly believe it’d be a miniscule bonus “per pack”. Given most of the foils I seem to get are common and barely worth anything. If you wanted to add a “rough” calculation you might be able to do something like this

1x Foil rare per 36 packs (take avg rare value x2 then divide by 36) and add this to your overall pack value.

3-4x Foil uncommons (take the avg uncommon value x2 then divide by 36 and multiply by 3 or 4)

But doing that you’ll see that the foil rare adds maybe 10 cents to a packs value. And the uncommons add about 5 cents…

yeah…One way to do it is to consider the first 35 packs as “normal” and the other one you add the foil value…but it was a lot of calculation just to add abit of EV. In my example, since lot of commons were “expensive” the added value was big enough to make a difference…the average price trippled and since you receive a foil each 4 packs…Well not exactly each 4 packs…see where I’m going…so I would do normal calculation for 3 packs, then add the foil value for 4th pack but since you can now double the common (normal +foil) it was well…a nightmare to still do an approximation.

in the end, if the Ev without foil is -EV well don’t count on foil to make it +EV and stay away from it. If it’s already +EV well things can just go better than expected because of foil!

Have one question: How was the land slot carried in DGM here?

WeQu came actually came up with the calculation to include the land, however, given the rate of shocks per box being around 1 and maze’s end being around 0.75 per box (though my own results are more like 0.33 per box), and my desire to keep it simplified, the land was not accounted for in the original equation. If you’d like to include the land spot you can add the probability of a shock 1/36 (or so)* the average value of a shock ($8.50) and you get another $0.24 per pack (for a rough estimate). However, I believe the main purpose of these types of calculations is to determine 1) if cracking packs is a worthwhile endeavor, 2) to compare various standard legal packs to determine which have the best EV (say if you have to choose prize packs), thus you’re sample quantity is likely to remain low enough that you’re unlikely to pull a shock or maze’s end…I like to consider this in the same boat as foil rare…they are possible, but unlikely in a small sample size.

You can see the calculations on the forum here.

To answer the “extra value due to land slot” question

– Maze’s End is a Mythic, so that means 1 in every 121 packs

– We know that the shocks are 1/4 as likely to end up (said by WOTC) and there were 2×5 on the print run for RTR/GTC (~6/box), so I believe there are 2 print runs with each 5 shocks, leading to 10/242 (~1.5/box)

I used the following formula to calculate the extra land slot value:

1/121 x Maze’s End + 1/242 x Total Shock Value

For today, this would lead according to TCGPlayer prices to

LOW = 1/121 x 0.50 + 1/242 x 74.15 ~ $0.31

MED = 1/121 x 1.64 + 1/242 x 94.46 ~ $0.40

If there are more questions, drop them in the forum 😉

I think it’s likely an in print set will be opened until it is slightly below $90 per box in expected value. This follows directly from the demand for singles driving the demand for packs. The big question is how much drafting can drive the EV of a box… A good corollary to this is the mean value of a set should tend towards a constant value as long as the set is in print. This means a high demand mythic (Jace, voice) can suppress the value of other relevant cards in the set!

Great work. this really shows that unless your getting a really good deaI per booster there just not worth it.

confess I love opening packs but hate sorting. so for dragons maze I’ve only bought one box and lucked out with voice,ral,2 shocks, and 4 other mythics.

Nice box…and ya, that’s the idea. It’s typically why more competitive players don’t buy boxes (except maybe pre-orders).

great article dave..could you do the ev of a zendikar box

Zendikar (as zendikar doesn’t have too many highly desirably uncommons..and a lot of bulk ones I won’t factor them)

Avg Mythic-5.91

Avg Rare-3.91

Weighted average calculation;

0.125*5.91+0.875*3.91=$4.16

So your average rare value is $4.16 per pack of Zendikar. It’s important to remember that this does not include foils AND does not include the “Hidden Treasures” available in the first wave of product (should you have that available to you)

I was pretty excited about this line of thinking untill I started calculating myself, and found out those prices used are the bots selling prices. No way we can sell those cards for those prices, so these EV is unuseable. It would be nice to see those calculations made with bots buying prices, that will give a whole new picture.

Bots?? Sorry my focus was on paper magic, not MTGO. I believe Sebastian’s article was EV with regards to MTGO, though I don’t think he did a per pack analysis. If you’re wanting to do an EV per pack based on buylist (or in your case bot buy prices) you’re pretty much NEVER going to find +EV, otherwise the bots/store owners would just crack all the packs themselves.