## Poker Math How-To: SB vs BB short-stack

It's folded to you in the small blind. Your villain has 20 big blinds left after posting the blinds. You look down at K5o. Can you show him your hand and shove over him and be assured that it's a profitable play?

Step 1: What range of hands can villain call with profitably if we shove K5o?

Villain will be calling 20bb in a pot of 21.5bb, so he can call profitably with any hand that has 48.2% equity or better against K5o. This range of hands is: 22+,A2s+,K5s+,JTs,T9s,A2o+,K5o+.

Step 2: If villain calls with exactly the range from Step 1, are we profitable?

Villain's optimal range found in step 1 is 30.6% of hands. It also has 66.0% equity against K5o. The value of our shove is found by adding up the following:

We shove and he folds (profit of 1.5bb): (1-0.306) * (1.5)
We shove, he calls, we win (profit of 21.5bb): (0.306) * (1-0.660) * (21.5)
We shove, he calls, we lose (profit of -20bb): (0.306) * (0.660) * (-20)

Which comes out to be (1.041 + 2.237 - 4.039) = -0.761. This means that if villain calls with the optimal range against our hand that we will lose on average 0.761 big blinds per hand.

Now a question for you to work out and answer: What is the range of hands you can just shove here and be guaranteed to profit, even if your opponent could see your hand?