Pocket Pairs in the Big Blind

[Brom]

I have a math question involving pocket pairs in the big blind. It's mostly for lowish pocket pairs where one would have to hit a set to win the pot, but not really exclusive to just low pairs.

First I'm going to lay down a few assumptions to follow that make this problem a little more simple. I'm going to assume that the odds of one hitting a set or better on the flop are 7.5:1, and that when one hits this set (or better), it will win every time. Second I'm going to assume that there is a 0% chance of a limp reraise in the scenario, and there is a 100% chance that the players who called one bet preflop will call one more. And lastly, I am assuming a standard limit Hold'em game with a small blind half the size of the big bling, the big blind equal to one small bet (on the preflop and flop betting rounds), and the big bet equal to double the small bet (on the turn and river betting rounds).

Now for the scenario question. Say I'm in the big blind holding 5/images/graemlins/spade.gif5/images/graemlins/heart.gif (note the suits are unimportant), three players have limped in, the small blind has completed, and the action is on me. Should I raise this knowing that the one small bet I commit to the pot here is going to build a pot that is 9 small bets (2 bets from each of the four other players plus one from my big blind)?

I usually don't raise low or mid pocket pairs out of the blinds with only a few limpers in, but can I do it for value? My one bet seems to be collecting 9:1 odds when I only need 7.5:1 to hit a great hand. Maybe this one has been discussed before somewhere and I just couldn't find it. If it has and someone knows of it could they post the link or reference the book where it was written? Am I totally off on this play in thinking that it is profitable, or have I been missing out on a lot of profit in the past?

[jdl22]

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I usually don't raise low or mid pocket pairs out of the blinds with only a few limpers in, but can I do it for value? My one bet seems to be collecting 9:1 odds when I only need 7.5:1 to hit a great hand. Maybe this one has been discussed before somewhere and I just couldn't find it. If it has and someone knows of it could they post the link or reference the book where it was written? Am I totally off on this play in thinking that it is profitable, or have I been missing out on a lot of profit in the past?

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Your thinking is off here. You are saying that you're in the bb and there are 4 other players in the pot. If you raise you will only be getting 4:1 on your money, not 9:1.

This and other similar types of errors are common. The problem is that when you are considering raising versus calling or checking you need to consider how much more money is being put in not the total amount that will be in there. So if you check the pot will have 5 bets in it while if you raise it will have 9 not counting the additional one you put in. That's a total of 4 bets added to the pot for your 1 put in so you are getting 4:1 on the raise. If UTG raised and the others called you would be getting 9:1 on the call. If you raised in that case you would still only get 4:1 because the difference between calling and raising is 4 bets and you put in 1 extra.

What you have done is demonstrate that raising is +EV. That is one step in the process of determining what should be done but isn't the last step. What we care about is not whether a play is +EV but whether it is the one that maximises your EV for the hand. Let's assume that raising doesn't affect how well you get paid off after the flop so that part can be ignored.

EV of checking:
odds against hitting a set on the flop are 15:2 so if you play the hand 17 times you will twice win. The pot contains 5 bets so your EV is 10/17.

EV of raising:
This time you win 9 bets the 2 times you flop your set but unfortunately you lose 1 bet the 15 times you don't. That makes the EV of raising (1/17)(18-15)=3/17.

Here you can see that the EV of calling > EV of raising. That's because the return you are getting from raising (4:1) is not as good as the odds against hitting your hand (7.5:1).