I bet 20 into a 30 pot on the turn so it has to be succesful 40% of the time to show a profit without equity. I know that it will only be succesful around 30% of the time but I think with a river barrel on certain cards it could show a profit.

My plan on the river is to bet if it's an A,K,J,9,8,7 and check if it is the rest. That's 24 combinations out of the 46 remaining cards, roughly 50%

So I bet on the turn 20 and win the pot immediately 30% of the time. I check on the river 50% of the time and lose the pot. I bet on the river 35 50% of the time, and win the pot X% of the time.

Here is the question: How much do my opponent needs to fold to make this play profitable? Or what is the minimum percent for X to make it a profitable play?

This is the equation I came up with:

(0.3*30)-(0.7*0.5*20)+(0.7*0.5*X*50)-[0.7*0.5*(1-x)*55]=0

win the pot on the turn 30% of the time: 0.3*30

bet the turn, check the river 50% of the time, lose the pot: -0.7*0.5*20

bet the turn, bet the river, opponent folds and I win: 0.7*0.5*X*50. <--- do I need to calculate with the 50 that my opponent put into the pot, or with my turn bet also, so 70?

bet the turn, bet the river, opponent calls and I lose: 0.7*0.5*(1-x)*55

I think, but not sure, you need to win at least "5" as summary of all river actions.

5 because you have to complete turn FE breakeven. If you need to win on Turn 40% with a 20 bet size and realize only 30% you miss again 25% of success. [30%/40% = 75% realization; 25% of 20 is 5]

I think x=56%

Looks right to me, use 50, not 70, b/c your turn bet isn't EV you are gaining. (assuming starting stacks of 55, if you get villain to fold on the river your stack will be 55+the pot of 70 - the 20 you put in on the turn = 105, which is a gain of 50).

I got X= ~46.95% using the 50% river bet, using a river bet of 24/46 (~52.18%), it's only slightly lower at ~46.37%.