A Little Math: Fold Equity

Q: You're on the button in a $1/2 NL cash game with full 100bb stacks. The action folds to the player in the cut-off, who open raises to $6. You know he has a very wide range here. You decide to 3bet (re-raise) to $18 and see if you can just take down the pot (i.e,. bluff). Both blinds muck their hands. Ignoring the rake, how often does the villain have to fold to your 3bet to make this a profitable play?

A: Okay, first things first: We have not said what our actual hand is. If the villain calls (or reraises), we would have to factor in our hand strength equity against his range, but that's not what the question asked. We're simply trying to calculate how often we need the villain to fold to take down the pot such that we show a profit over the long run with a bluff here. In other words: How much fold equity do we need to make this a plus EV play? Here's how it's calculated:

You're risking $18 to win $9 (i.e., the villain's $6, plus the $3 in dead blind money). 

Said another way, you're giving yourself odds of 9:18, or 1:2.

This means that to break even we need to win 2/(1+2) = 2/3 = 66.6%

If the villain folds at least 2 out of every 3 times in this situation, the play is +EV and you should always re-raise with any two cards in this situation.

All-in for now...

-Bug

PS. Note that if the villain folds less than two out of every three times, this may still be a profitable play to make. Our hand will have some equity against his range and might in fact win outright at a showdown. We'll also we'll have position, which gives us an edge to try to out play him on the flop, turn, and/or river.