The Orrery: TES:L Arena Curves
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RarityCountsPercentage*
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Common103161.37%
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Rare43625.95%
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Epic17410.36%
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Legend392.32%
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Total1680100.00%
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* This worksheet is in a constant state of being updated to better represent the probabilities involved. Percentage calculations are an estimate, and not final.
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Notes on calculation:
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Rarity Percentage:
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The above counts are used to calculate the actual probability that a card of a certain cost will appear, as shown in the 'Weighted' table. Counts in the 'Rarity' table are taken from my personal arena runs beginning 1/6/2017 (rank 1 Versus Arena only, Solo Arena and Chaos Arena counts are not utilized). The probability that a certain rarity will appear is given by Counts/Total for that rarity. For example, if Epic 'Counts' = 20 and Total = 210, probability of an epic pick being offered will be 20/210 = 9.52%. Note that these values will consistently update as I add counts from additional arenas.
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% by cost:
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The '% by cost' values represent the probability that a single card at a specific rarity will be a creature of a specific cost. This value is calculated by dividing the number of available creatures at a cost and rarity by the total number of cards available at that rarity for the given attribute/class. For example, there are 8 3-cost agility cards at common and 23 total agility commons. So '% by cost' = 8/23 = 34.78%. '% by cost' values for classes are calculated using card counts for both its attributes as well as neutral cards.
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Single Card Probability:
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This value shows the probability that any single card that appears as an arena pick will be a creature of a specific cost, weighted by rarity. It is determined by multiplying the '% by cost' probability for a specific cost at each rarity by the rarity probability shown in the 'Rarity' table. Thus the equation to calculate single card probability (SC%) for a class Y at cost X is as follows, where p indicates '% by cost', and C, R, U, and L indicate rarity: SC% = (YXpC * Cpercentage) + (YXpR * Rpercentage) + (YXpE * Epercentage) + (YXpL * Lpercentage) Note that this does not indicate the total probability you'll be offered a creature of a certain cost as a potential pick (see Three Card Probability).
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Three Card Probability:
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This value shows the probability that you will be offered a creature of a certain cost as one of the three cards available to you as an arena pick. Note that this value is not equivalent to three times the single card probability. The equation to calculate three card probability (TC%) is as follows: TC% = (Cpercentage * (1 - (1-YXpC)^3)) + (Rpercentage * (1 - (1-YXpR)^3)) + (Epercentage * (1 - (1-YXpE)^3)) + (Lpercentage * (1 - (1-YXpL)^3)). Why this equation? (1 - YXpRarity) indicates the probability that a single card at a rarity will NOT be of the specified cost. The probability that all three cards will not be of the specified cost is this value multiplied by itself two more times: (1 - YXpRarity)(1 - YXpRarity)(1 - YXpRarity), or (1 - YXpRarity)^3. Once we've calculated the total probability that a card of the specified cost at a rarity will not appear, we can subtract it from 1 to calculate the percentage that it will appear at that rarity. We multiple this percentage by the percentage that a specific rarity will appear: Raritypercentage * (1 - (1-YXpRarity)^3) The sum of the percentages for each rarity gives us the total probability that at least one card at a rarity will appear.
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Note: the values in these tables assume that there are no additional weighting criteria involved in determining what area picks are offered other than rarity (e.g., different probabilities varying by arena rank, promotional cards/cards from specific sets being offered more or less frequently as in Hearthstone)
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