Expensive Free Cards

[AngelicPenguin]

[ QUOTE ]
Now, assume the turn is a 9spade. The pre-flop raiser bets, and the loose-passive opponent folds. This would be a good time to raise as a semi-bluff. Assuming you have nine clean outs, if your opponent folds ten percent of the time, raising is more profitable than calling.

[/ QUOTE ]

Does anyone know how to do the math behind this?

Here was my feeble attempt, making some assumptions.

We're getting 5.1 to 1 on the turn.
So in the raise scenario
% of the times villain folds * pot we win + % of the times he doesn't fold (pot if we improve - cost of raising the turn)
.10 * 5.5 + .9( 9/48 * 10.5 -39/48 * 2 ) = .89BB
So 10% of the time we win the pot outright, and 90% of the time we either get to raise the river when our flush comes in or we fold. I'm pretty sure this math is wrong. Don't you normally have to do "1-" the first scenario or something like that?

The second scenario is probably even more screwed up.
We call.
Chance we improve on the river and get to raise - the cost of the call on the turn.
9/48 * 10.5 - 39/48 * 1 = 1.157 BB

So is my math wrong (probably), or my assumptions wrong (probably) or both? I didn't want to overcomplicate the formulas by including when he would check the river when the draw comes in and we only make one bet, the times he 3-bets the turn with a monster, etc.

-Matthew

[raisins]

Here's the way I did it.

7 1/2 small bets pre flop. Another 3 on the flop. 5.2 (or a bit less with rake) bets in on the turn. You are getting 6.2 to 1 on your action. You make the flush about 20% of the time.

No semi-bluff: 20% of the time you win a 7.2 big bet pot + 1 on the end, 8.2. 80% of the time you lose one big bet.

Semi-bluff: 20% of the time he calls your raise you win a 9.2 big bet pot (5.2 + 4 bets on the turn) + 1 on the end, 10.2. 80% of the time he calls you lose two big bets. When he doesn't call your turn raise you win a 7.2 big bet pot.

So, the math question is, how often does he have to fold to your turn raise for these two plans to break even?

x<=1

(8.2)(.2)+(-1)(.8)=(x)(7.2)+(1-x)[(10.2)(.2)+(-2)(.8)]
.44=6.8x
x=.065=6.5%

So, why does the author have 10%? I don't know, man, I didn't write it. Probably because he rounded up and you should too. Err on the conservative side when coming up with these estimates as there's lots of things that will go wrong that the formulas don't take into account. Some of the unaccounted bad possibilities include, villain is fast playing a K high flush draw, he doesn't pay off the river, he actually had a set and your flush card boated him up, he 3-bets the turn, plus you actually make the flush a bit less than 20% unless he holds no clubs. 10% seems like a good safe estimate.

Looking over your math I see that the size of the pot hero wins when the semi-bluff raise is successful is too small, also the number of unknown cards is too large making the chance of completion too small.

Let me know if this helps.

regards,

raisins

[raisins]

I was thinking about this hand a bit and there's a math error in my post. When the semi-bluff works you win an 8.2 BB pot not 7.2. This makes the solution to the equation .056. So the estimate is a bit more conservative then previously stated.

raisins

[stoxtrader]

nice work raisins. thanks.

the salient point, I hope, is that taking a "free-card" in a heads-up or 3 handed pot is often inferior to continueing a semi-bluff and then simply firing a 3rd shot on the river improved or not.