Oh you're not being an ass at all. Thanks for bringing this up, I appreciate experimental evidence. TBH I felt it was high too, but I trust my math more than my intuition. As I said, "I still might have made a mistake". However, please read the fine print in my message: PST = Probability of Second Turn So none of those percentages actually end up being 38%... (Unless you always have second turn) Assuming PST = 50% then you get No mulligans : 16.96% No Free Mulligans : 19.02% One Free Mulligan : 29.525% To address the mathematics issue: First, your notion of the required cards is wrong, for the combo you need Caverns, Flash, Spirit or Summoner's, Hulk or Summoner's Both Spirit and Hulk are interchangeable with Summoner's as it puts it in your hand Since statistically speaking in a randomly shuffled 60 card deck the probability of you drawing one card out of a set of 4 in 7 draws without replacement should be roughly 40% (no mulligans) I would think that you aren't using 4 Gemstone Caverns. Or your sample size is too small. Onto the calculations: After reviewing it in light of your experimental evidence it seems I did quite a bit of double counting by letting the rest of the cards be free cards and then permuting i.e. when your opening hand has 2 Hulks in it or 1 Sum. and 1 Hulk after assuming there were 0. If I have spare time I'll redo the calculations. This will probably be some time next weekend. Rough estimate though? Around 4-8% * PST. We'll see how close this estimate is I guess.
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I don't know why everyone is attacking you. Yes your math is off, but it isn't like probability is the easiest subject. And to by honest you were on the right track... But unfortunately, the people mocking you know a little but are still falling short... To get a 0 turn win you need to have the right cards and not have 3 Disciples in your hand. Probability of drawing in n cards (in any order) w/o Disciple (Caverns)(Simian)(Hulk)(Flash)(Cards x n-4) (4/60)(4/59)(4/58)(4/57)(n!) [ if n = 7 then multiply by 52/56 to force 1 card to NOT be Disciple ] or with 0 Simians and at least 1 Hulk (Disjoint from above) (Caverns)(Sum. Pact)(Hulk)(Flash)(Cards x n-4) (4/56)(3/55)(4/54)(4/53)(n!) [ if n = 7 then multiply by 48/52 to force 1 card to NOT be Disciple ] or 0 Hulks and at least 1 Simian (Disjoint from above) (Caverns)(Simian)(Sum. Pact)(Flash)(Cards x n-4) (4/56)(4/55)(3/54)(4/53)(n!) [ if n = 7 then multiply by 48/52 to force 1 card to NOT be Disciple ] or 0 Simians and Hulks (Disjoint from above) (Caverns)(Sum. Pact)(Sum. Pact)(Flash)(Cards x n-4) (4/52)(3/51)(2/50)(4/49)(n! / 2!) [ if n = 7 then multiply by 44/48 to force 1 card to NOT be Disciple ] Let PST be the probability of getting the second turn ( In general this is not 50/50 ) Then the probability of winning first turn comes out to be, drum roll please: No mulligans : 33.92% * PST No Free Mulligans : 38.04% * PST One Free Mulligan : 59.05% * PST So assuming a 50/50 split (and thorough shuffling) you can win ~30% of the time if you are allowed a free mulligan. Granted, although I aced the MFAT in mathematics I still might have made a mistake. If you don't believe the numbers feel free to double check them. However, I'm fairly confident in them.