we are opening from the button hoping the BB and SB fold. theres antes in play which add up to 1BB. so a total pot of 2.5BBs.

How often does a button open have to work to return an immediate profit?

Open to 2BB: 44%
Open to 2.25BB: 47%
Open to 2.5BB: 50%
Open to 3BB: 55%

are these numbers correct? how do you work out these numbers?

thanks.

Returning an immediate profit with a button open means that both blinds fold. The needed fold probability (fe) is gotten from the equation:

EV = fe*Pot –(1-fe)* Bet,

where fe is fold equity when you make a bet = Bet into a pot = Pot. This equation is based on the assumption that if at least on blind calls, you lose - i.e., you have an air-hand.

Set EV to be >= 0 to establish the required minimum fold equity and you have

fe >= Bet/(Pot+Bet),

which correspond to the numbers you showed.

Note that Bet/(Pot + Bet) has been termed alpha and 1-alpha is the minimum defense frequency (MDF). Also note that for this example with both blinds having yet to act, if they act independently (probably not true) the individual folding probabilities is the square root of alpha. Thus, for a pot size bet, to win with a pure bluff, total needed fold equity = 50% so each blind will have to fold with probability of about 70%.