I posted this on a Magic forum and got zero responses.
I'm playing a High Tide combo deck. Let's say I am Brainstorming with 50 cards in my deck, 23 of which are outs that make me win. I think the calculation for this is:
(27/50)*(26/49)*(25/48), so my failure rate is just about 15%.
However, drawing Ponder, Brainstorm, or Preordain off of my first Brainstorm might enable me to dig for an out. I only have enough mana to cast one more cantrip and one out, I have 3 Brainstorms, 4 Ponders, and 3 Preordains left in my deck. I came up with this:
((27/50)*(26/49)*(25/48))*((47/50*46/49*45/48)). (I think) This would be my percent chance of not hitting one of my 23 outs and not hitting a brainstorm, around 12.3%.
Of course just drawing Brainstorm isn't good enough, I need the third card from the top of my deck to be one of my 23 outs (the top 2 are cards I replaced from brainstorm). It is here that I'm stuck, I don't know how to figure that out, or how to figure it out for Preordain or Ponder. It's been a while since I've taken stats, I don't even know if that 12.3 is correct.