For casual observation of patterns with a fairly large chance of statistical error? a few hundred deck switches (I switch decks every 3-4 games, if not more, so my playstyle is actually conducive to casual testing of this)
For low statistical error? you'd want several thousand deck switches.
I can't provide a proof that would undeniably prove that the RNG is broken, but I can provide "better than anecdotal evidence" with my sample set.
As an aside, if I were to test the RNG right now, some awkward results in-game recently would actually lead me to wanting to test "CotC Only" vs "CotC/DP Hybrid" vs "DotP only" test sets in addition to just the RNG. I would recommend this for anyone looking to provide a statistical proof for it.
Some criteria I would set forth if I were to set up a test set:
Create 3x "CotC" decks, 3x "Hybrid" decks, and 3x "DP" Decks. each using 10 cards (4 of each) - 41 card deck w/ hero. (Hybrid deck would use 5/5 split)
Initial Test - Switch decks between each game, enter 99 games (rotating to the next deck for each game) - track opening hand and first 3 turns of draw.
Example:
Game 1 - CotC Deck 1
Game 2 - CotC Deck 2
Game 3 - CotC Deck 3
Game 4 - CotC Deck 1
Game 5 - CotC Deck 2
Game 6 - CotC Deck 3
etc.
(this would result in 33 tests with each deck)
Indicate frequency of "all 1 of", "2 of" , and "3 of" in opening hand. Same for successive draws. - Statistically speaking 2 of is reasonable, a high rate of 3 ofs would be problematic.
After this, enter 33 games with each deck without a rotation between, and track these results as well for a comparison.
(there would still be a high probability of error, but this sample set could show leanings, especially if the leanings are statistically 'large')
You could also do a large-format test using a Human "All Cards" and a Shadow "All Cards" and flip between the two to get a different statistical set. (could isolate CotC, DP, and Hybrid setups as well)
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