In the video game Hearthstone, you acquire card by opening packs. Each pack contains 5 cards of varying rarities: common, rare, epic and legendary. There is a different percentage chance of a card being of each rarity. A pack can contain any combination of rarities except 5 commons.

I'd like to calculate the expected value of a pack using the game's currency "Arcane Dust," which I'll refer to as "Dust" for the remainder of the post.

If I do not have the card, then the card is worth the full value for the rarity. If I do have the card, it's worth its D/E (disenchant) value.

I'm trying to figure out what should go in the "?" cell. Ignore the empty cell at the bottom of the D/E Value column.

Table Explanation: The yellow colored cells contain data on the chance of a card being of each rarity, as well as their full and D/E values. The red cells contain, in order, the total amount of card of that rarity, how many are missing from my collection, and the expected value of a card of that rarity. For example, the expected Dust value for a Rare card is

$$\frac5{162} \times 100 + \frac{162-5}{162} \times 40 = 22.47$$

I can calculate the expected Dust value for a single card by taking a weighted average of the values in the column on the far right, which comes out to 6.32. Multiplying this value by 5 doesn't work, because we are guaranteed not to get 5 commons. I'm really struggling figuring out how to include that requirement in the formula for expected Dust value of a pack.

Given this information and the table, what is the formula for calculating the expected value of a pack?


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