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My take:
I agree 100% with Max and Marshall.
BB's best card is a 7. BU would likely check behind any 7's he had on the flop while BB would almost surely call them all. I'd say second place is an 8x.
BU's best card is clearly a Ks or a Qs. Completing the flush draw is better for BU simply because BU has more suited cards in his range preflop than BB. The board is very important here. While we expect BB to have suited connectors and possibly gappers in his range, BU could have something like Kxs. Qxs, J8s, etc. On the other hand, its much harder for a suited connector to hit this flop (he can't have 76ss,87ss, T9ss, etc. Because of this, a Ts7c3s flop is better for BB). And because its possible BB could raise the flop with a fd, while BU is always betting them. In other words, BB has less flushes in his range than BU.
Explaining why other answers are wrong:
1. Ax or s best for BU: An ace is actually a bad card for BU, because he will check behind some ace highs, as well as A7/A3, so his cbetting range doesnt have many aces in them. On the other hand, BB has a lot of aces highs A7/A3 that he check/calls the flop with. An Ace may in fact be the 2nd best card for BB, its close between an Ace, 8, or 9.
2. T is the best card for BB: I'd say this is a neutral card. BB has just as many T's as BU. But a factor not touched yet is that a T does not significantly increase the chances that BB will win the hand. If a T comes on the turn and BB has T9 he is now beating only four more hands, AA-JJ. On the other hand, if a 7 came, BB would now be beating 88/99, all T's, and AA/JJ.
Marshalls post touches on another factor no one talked about: How do turn cards affect how passive or aggressive either player will be. If BB feels he is ahead of BU range, he will want him to be more aggro on the turn. If BB feels he is behind BU's range, he will want BU to be more passive on the turn.
So what?: Here's some information you probably didn't know was true; if I'm button I am never double barrell bluffing on a 7. The only way I'd double barrel a 7 is if BB always leads his 7x's if he hits a 7 on the turn. Here's some information you also probably didn't know is true; I double barrel like a monkey on Ks turns. Why? Because unless BB wants to make a hero call or has KT, he is going to have to make a hero call on the turn and possibly river. Also, if an observer with incomplete information on both BB and BU's hands tried to weigh the % chance of BU and BB winning the hand, a Ks would increase BU's chances the most, while a 7 would increase BB's chances the most.
What did we learn:
Concept: Build pots when you are ahead of an opponents range, keep pots small when you are behind an opponents range.
Expression of concept: Double barrel cards that statistically increase your chance of winning the hand (strengthen your range), check behinds cards that statistically decrease your chance of winning the hand (bad for your range)
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