I am of the firm belief that raising for a free card, a.k.a. "the free card play" is the single most important "move" you can make in low limit online poker. This is a textbook case, and if you have any questions, I'd be happy to help.
Party Poker 0.5/1 Hold'em (10 handed) converter
Preflop: BMF'er is BB with [7d], [Ad].
7 folds, BMF'er calls, Button calls.
Position is important. The big blind isn't great, but it is when everyone folds to the button. I put the button on a button-wh0re raise, and the SB on a resteal (maybe a semibluff small pocket pair or something). Either way, I don't mind taking a flop for three bets with a suited ace one in front of the button.
Flop: (9 SB) [8d], [6d], [Jh] (3 players)
SB bets, Button folds, SB calls.
...And that's why. For the preflop action, you can assume SOMEbody's going to be autobetting the flop, probably the guy who got the last bet in preflop (who we're conveniently behind).
I raise here for a free card on the turn. I have the nut flush draw and an overcard that may or may not be 3 more good outs. For all the action we've seen so far, unless he's sitting on a pretty big hand, he probably won't send it 3 back to me, and also will check the turn to see what I do.
This is why if you're not making the free card play, you don't like money. Think binomially. I will either hit my flush on the turn or river, or I won't. If I hit the turn, I have the nuts for a big betting street, and the probable nuts on the river. Raising makes my winning pot bigger. If I miss the turn, but hit the river, then the turn goes check-check. Had I called, I'd have spent 1.5 big bets, instead of 2 small bets.
Turn: (6.50 BB) [Kh] (2 players)
SB checks, BMF'er checks.
Swingandamiss. Check/check, and the joke's on him. Shorthanded, I could have even bet this turn for fold equity alone.
River: (6.50 BB) [Td] (2 players)
SB bets, SB calls.
...And there's my beautiful card. Nuts good, and I even stole another bet from him when he spewed into me.
Final Pot: 10.50 BB
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Now lets go into a comparison of what happens calling vs. "free carding." Say you have the nut flush draw on the button, and there's a bet to you, and 4 people including you see the river.
Lets say you call the flop and hit on the turn. You should be able to get in a raise and hopefully a river call, for +12-16 big bets.
Let's say you call the flop, miss and call the turn, and hit the river. Either 10-14, depending on if he leads into the river.
Let's say you call the flop, and miss the turn and river. You're potentially -1.5 big bets (folding ace high to the river bet).
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Now let's say you raise the flop, and hit the turn. Now you're getting 10 bets, and your money goes in when you have the winner.
If you raise the flop, miss the turn, and hit the river, then you got that money card for free. And you're still making 8 bets, and as much as 12 if he leads into you on the river (like what happened in my situation).
And, most of all, you get out the cheapest if you miss, for 1 bet total instead of 1.5. Plus, you get to instill the fear of your raises when you hit, and people that don't know poker think you're a loose money huckle who bets/raises on the draw.
I realize most of the people that post and read strategy are well familiar with this. But for the people who aren't, and the lurkers that are out there, I hope you learned something.