I'm pretty good at math but this particular one I can't figure out. Probability is so annoying.

Let's say a set has 5 mythics worth $15,$16,$17,$18, and $19
and it has 10 rares worth $1,$2,$3,$4,$5,$6,$7,$8,$9, and $10

If I want to calculate the estimated value of the average booster pack (disregarding foils and uncommons)
The average mythic cost would be $17 and the average rare would be $5.50. So since mythics are printed at half the amount on the sheet as rares, so the equation should be
(17 + (5.50 x 2)) / 3 so that rares are weighted double. So that'd be $9.33 average return per booster.

Except if I take the sequence of numbers 15,16,17,18,19,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10 and average it, I get $7.80.

Both methods appear to weigh rares as double and both seem correct. Obviously the difference in numbers is because there are 10 rares and 5 mythics and it would be the same number if there were 5 rares. So that's great except which method of calculating results in the actual, true expected value of a pack? It seems to come down to what WOTC actually means by mythics are twice as rare as rares. Do they mean for every Force of Will in eternal masters, there are two Isochron scepters? Or do they mean that for every general mythic slot on the sheet there are two general rare slots regardless of the individual card. I'm leaning towards the first one from what I know about sheet printing but please, someone that's good at this type of thing let me know

For any one mythic in a set there is two of any one rare, this is why mythics appear at an average of 1 in 8 packs. If it was a simple 1 to 2 slot then mythics would have to show up 1 in 3 packs. The problem with your numbers is that you don't have the right balance of rares to mythics which throws off the math.

For any one mythic in a set there is two of any one rare, this is why mythics appear at an average of 1 in 8 packs.

This combined with the fact that they design for about 3½ times as many distinct cards at rare as they put at mythic (small sets' 35 to 10 = 3.5 exactly; large sets' 53 to 15 is a 3.533:1 ratio but has the nice result that all the cards mesh neatly into a full 11x11 sheet for the printing machines) is what yields the emergent result of "1 in 8".

Remember that it doesnt take into accounts:
-the prices that are unstable
-your ability to sell the cards
-your price average results in pulling out the perfect odds of what has been establish by MTG. If you buy 2 boxes, the price average may differ alots (could be as high as 80%) from your expectation while if you buy 10000 boxes, the price average will be pretty close to your expectation.

I'm pretty good at math but this particular one I can't figure out. Probability is so annoying.

Let's say a set has 5 mythics worth $15,$16,$17,$18, and $19
and it has 10 rares worth $1,$2,$3,$4,$5,$6,$7,$8,$9, and $10

If I want to calculate the estimated value of the average booster pack (disregarding foils and uncommons)
The average mythic cost would be $17 and the average rare would be $5.50. So since mythics are printed at half the amount on the sheet as rares, so the equation should be
(17 + (5.50 x 2)) / 3 so that rares are weighted double. So that'd be $9.33 average return per booster.

Except if I take the sequence of numbers 15,16,17,18,19,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10 and average it, I get $7.80.

Both methods appear to weigh rares as double and both seem correct. Obviously the difference in numbers is because there are 10 rares and 5 mythics and it would be the same number if there were 5 rares. So that's great except which method of calculating results in the actual, true expected value of a pack? It seems to come down to what WOTC actually means by mythics are twice as rare as rares. Do they mean for every Force of Will in eternal masters, there are two Isochron scepters? Or do they mean that for every general mythic slot on the sheet there are two general rare slots regardless of the individual card. I'm leaning towards the first one from what I know about sheet printing but please, someone that's good at this type of thing let me know

For any one mythic in a set there is two of any one rare, this is why mythics appear at an average of 1 in 8 packs. If it was a simple 1 to 2 slot then mythics would have to show up 1 in 3 packs. The problem with your numbers is that you don't have the right balance of rares to mythics which throws off the math.

This.

Your second method is correct, and your first method is not. When you take the average value of a mythic rare and the average value of a rare and want to calculate expected value, you need to weight by their relative frequencies of appearance. Therefore, simply dividing by 3 is not correct. In your first calculation, you're assuming that there are twice as many rares as mythics, which is incorrect. You example has 5 mythics and 20 rares, meaning that there are actually four times as many rares as mythics.

In order to use your first method, your equation should be (17 + (5.50 x 4))/5, which yields the same result as your second calculation: 7.80.

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I think you are over complicating it. The formula would be:

(1/8) * (Average Mythic) + (7/8) * (Average Rare) = Estimated Value of Rare Slot in Booster Pack

So you would get

(1/8) * ($17) + (7/8) * ($5.5) = $6.94

For the way Wizards puts sets together this is correct. For the hypothetical set in the OP (if there really are exactly 5 mythics and 10 rares in the set) it's a little bit off.

It's also a little bit off for any set that uses the standard "53 rares & 15 mythics" pattern that most large sets use. The actual ratios are 15:121 and 106:121, but everyone rounds them to 1:8 and 7:8 for simplicity's sake.

Let's say a set has 5 mythics worth $15,$16,$17,$18, and $19

and it has 10 rares worth $1,$2,$3,$4,$5,$6,$7,$8,$9, and $10

If I want to calculate the estimated value of the average booster pack (disregarding foils and uncommons)

The average mythic cost would be $17 and the average rare would be $5.50. So since mythics are printed at half the amount on the sheet as rares, so the equation should be

(17 + (5.50 x 2)) / 3 so that rares are weighted double. So that'd be $9.33 average return per booster.

Except if I take the sequence of numbers

15,16,17,18,19,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10and average it, I get $7.80.Both methods appear to weigh rares as double and both seem correct. Obviously the difference in numbers is because there are 10 rares and 5 mythics and it would be the same number if there were 5 rares. So that's great except which method of calculating results in the actual, true expected value of a pack? It seems to come down to what WOTC actually means by mythics are twice as rare as rares. Do they mean for every Force of Will in eternal masters, there are two Isochron scepters? Or do they mean that for every general mythic slot on the sheet there are two general rare slots regardless of the individual card. I'm leaning towards the first one from what I know about sheet printing but please, someone that's good at this type of thing let me know

This

combinedwith the fact that they design for about 3½ times as manydistinct cardsat rare as they put at mythic (small sets' 35 to 10 = 3.5 exactly; large sets' 53 to 15 is a 3.533:1 ratio but has the nice result that all the cards mesh neatly into a full 11x11 sheet for the printing machines) is what yields the emergent result of "1 in 8".(1/8) * (Average Mythic) + (7/8) * (Average Rare) = Estimated Value of Rare Slot in Booster Pack

So you would get

(1/8) * ($17) + (7/8) * ($5.5) = $6.94

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-the prices that are unstable

-your ability to sell the cards

-your price average results in pulling out the perfect odds of what has been establish by MTG. If you buy 2 boxes, the price average may differ alots (could be as high as 80%) from your expectation while if you buy 10000 boxes, the price average will be pretty close to your expectation.

Your second method is correct, and your first method is not. When you take the average value of a mythic rare and the average value of a rare and want to calculate expected value, you need to weight by their relative frequencies of appearance. Therefore, simply dividing by 3 is not correct. In your first calculation, you're assuming that there are twice as many rares as mythics, which is incorrect. You example has 5 mythics and 20 rares, meaning that there are actually four times as many rares as mythics.

In order to use your first method, your equation should be (17 + (5.50 x 4))/5, which yields the same result as your second calculation: 7.80.

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For the way Wizards puts sets together this is correct. For the hypothetical set in the OP (if there really are exactly 5 mythics and 10 rares in the set) it's a little bit off.

In your second example 1/5

In reality probability of mythic is 1/8

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