[TopThis]

[ QUOTE ]
Here's a hint. In almost all cases the ace-queen will check the flop and the player holding the jacks will bet probably somewhere between half the pot to the whole pot.

[/ QUOTE ]

Here is what I think Mason is getting at. Since we're assuming the JJ will always bet when checked to, when your AQ hits and you check, you'll win more than JJ will when your AQ misses and you check fold.

With JJ, an overcard comes on the flop 57% IIRC. Let's look at a few flop scenarios of JJ vs AQ. (ignoring the flush possibilities)

All undercards: AQ check folds.

An ace flops. Unless your opponent has AK you're good here. Check and see what opponent does. We're assuming JJ bets, so repop it with AQ and you take down a larger pot.

A King flops: AQ checks, JJ bets. Check raise here also and represent AK. Unless villain has AA here, you probably take it down. KK would not play this way, and QQ and JJ fold.
A Queen flops: This is probably what I don't want to see since KK is a likely holding for villain given preflop play.

Just taking a stab at what the hint might have meant? I realize this is an over simplification of the situation.

I'd definitely take JJ though also.