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I know they are found in the Underground while mining

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Do you mean Mysterious Shards? You cannot find the slates underground.

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The base rates are:

  • Mysterious Shard S: 30 in 1044 = ~2.87%
  • Mysterious Shard L: 10 in 1044 = ~0.96%
  • Either: 40 in 1044 = ~3.83%

The game can spawn two to four items (inclusive) in one session, so you could extend those probabilities to get an "in practice" drop rate, or even make an average. This might not give an accurate reflection of the game's real calculations, though.

Source

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Why does the game use a 1044 value?
1044 is the sum of the rates in the National Dex columns in the spreadsheet. There is no particular reason the rates add up to that number.