Answer 1: The way to do this is to figure up all of the possible combinations Villain opens assuming we were dealt A9s and all of the possible combinations that Villain continues with against our hand A9s. Those two numbers will give us a percentage of the time that Villain folds. Next, we figure how often we need Villain to fold to break even on our bluff. If Villain folds more than that, then it's obviously a +EV bluff.

So first we figure the number of Villain's opening combinations

Code:
Hand   Combos  Blockers   Total
-------------------------------
22+      78       6        72
ATs+     16       4        12
KTs+     12       0        12
QTs+      8       0         8
JTs       4       0         4
AJo+     36       9        27
KQo      12       0        12
-------------------------------
                          147
Now we figure the number of Villain's continuing combinations

Code:
Hand   Combos  Blockers   Total
-------------------------------
TT+      30       3        27      
AQs+      8       2         6
AKo      12       3         9
-------------------------------
                           42
The percent of the time that Villain folds is 1 - (42/147) = 0.7143 = 71.43%.

Since it's folded to us in the big blind and we are raising to 12x, we are putting another 11bb into a pot of 5.5bb. The percent of the time that Villain has to fold for us to break even is 11/(5.5+11) = 0.6667 = 66.67%. Villain folds more than that, so our 3-bet bluff with A9s is forced to be +EV regardless of any value we get post-flop.


Answer 2: The way to do this is to figure up all of the possible combinations that give Villain top pair or better on the Q94r flop. We already have the total combinations that make up {TT+, AQs+, AKo} from answering question 1 and we can just subtract 3 from that to compensate for the blocker Queen on the flop for QQ and subtract 1 from that to compensate for the same with AQs. Then the answer to question 2 is a simple percentage of those two numbers.

Villain's range is {TT+, AQs+, AKo} on a board of Q94r and the Queen on the flop is never the same suit as Villain's AQs to simplify the calculation a little. Villain hits top pair or better with QQ+, AQs. There are 3 combinations of QQ left (because there is a Queen on the flop), 6 combinations of KK left, and 3 combinations of AA left (because we hold an Ace).

However, the combinations of AQs are a little trickier -- there are only 2 combinations of AQs left. Remember: the Queen on the flop is NOT the same suit as our Ace. So if our Ace was Spades and the Queen on the flop was Clubs, that leaves only two possible combinations of AQs (Hearts and Diamonds).

So this leaves us with 3+6+3+2 = 14 combinations that hit top pair or better on the flop, and 14/38 = 36.8%, so Villain hits the flop 36.8% of the time.


Answer 3: This was by far the easiest question. Another way of asking this question would be 'Which starting hands in Hold'em hold the most effective blockers against the range {TT+, AQs+, AKo}?' There are a few different ways to do this, but I'll outline a way that's easy to learn.

First, consider each "part" of Villain's range. That is, consider TT+, AQs+, and AKo seperately. Now, ask yourself which hand(s) provide the most blockers for each?

a. For TT+, any non-pair hand where both cards are T or higher provide 6 combinations worth of blockers.

b. For AQs+, any *offsuit* AQ or AK provides 3 combinations worth of blockers. For example, if we hold AsQd, then this blocks the combinations AsQs, AdQd, and AsKs. (Note that a suited AQ or AK only blocks 2 combinations).

c. For AKo, any AK provides 5 combinations worth of blockers.

Now we can see that AKo provides the most effective blockers for each part of the range, so it will provide the most effective blockers against the entire range as well.


Now it's time for you to compare your answers to my own and ask questions.