You were lazy doing the math. Shauna is a Guaranteed 5. Birch is a 50-50 4-7. 50=1/2 100. 50 x 2=100. 4 x 2=8.50/100=1/2. 4/8=1/2.4/7<4/8. The math is against you with Birch, being that a 1/2 coin flip is getting you a <1/2 outcome. Mathematically, it's just better to play Shauna. Plus, I've played games where I decked out because I landed heads on Birch. So, with Judge, you get the same 100% determination that you do with Shauna, but the outcome of a Birch. Except for the fact your opponents loses cards from their hand. That all being said, Birch is only deserving of a 1-of in a deck, where you would play Shauna/ Judge in the other spaces. Placing more would decrease your odds even more, theoretically having worse outcomes because 50-50 coin flips times 4-7=200-350 x 4 would be 800-1400, which playing a 1-of would have 50-50 coin flips times 4-7 would just be 200-350. That, if I did the math right, would decrease your odds of bad coin flips by 4, basically forcing yourself to waste 2 turns of draw power within 1 turn by not playing Shauna. No matter what you say, a definite 5 over a <1/2 of getting 7 over 4( Like I said, is less than 1/2 of what your drawing for.), is pretty bad. It would be different if it gave you 1 more card, evening the gamble to 1/2, like it should be. Basically, the average doesn't matter in an uneven tradeoff. Because while you can mathematically do it, you can't physically, which you obviously forgot. And for T-mail and Acro Bike, Acro Bike burns through your deck faster, getting what he needs to draw into, mathematically giving them a better bet of what they need, because it burns 2 cards off of your deck, giving you a 1/2 chance of getting a card each time you play it. While physically you can't do that(Because you have a 100% of getting a card), you can at least increase your odds by 3.33% each time you play one. Playing 4 increases that to about 12.5. 12.5/60=4.8/32.5520833333 dividing that equals to 6.78168402778. So, that means that if you played 6.78(That rounds up to 7)Acro Bikes, you would almost guarantee you draw into something you need. Playing Shaymin and Trainers' Mail will make more than the difference of 3. Trainers' Mail, while technically, doesn't mathematically increase your odds(Because of the shuffling cards back into your deck), it can be helpful to shuffle, because if you didn't get it the first time, you have better odds with a shuffle, randomizing the deck again, generating random things, and each time you randomize, you have about a 1/3 chance of changing a card. Because you can't physically have those odds, it does physically round to about 3/4? Basically, playing 3 Trainers' Mail you will completely have different cards in the 3rd one than in the 1st one. Playing 4 increases those odds even more, being especially helpful in speed decks, such as this deck. Making maxing out both Acro Bike and Trainers' Mail highly optimal.
