Forgive me if I being thick. Also, I'm not trying to be antagonistic.
@ Oskar. I copied a solution from someone else as an example of a calculation I couldn't do whilst playing a hand.
@ Spoon. I do get the point you made earlier that you didn't intend this to be done during play.
I didn't randomly pick numbers. I took the ratio between amount that might be lost v the amount to be won. $2 : $5. ie 40 : 100. He has to fold 40% of the time for this to be a good play for us of we exclude the four outs. The quick calculation for outs is 2% per out per street. Four outs = 8% chance. An 8% improvement on the 40% (in absolute terms, not relative). This means he must fold 32% of the time for our turn bet to succeed. I thought this seemed a reasonable (pretty close to 31.7%) way to answer your original question in the time permitted during a game. I'll be quite happy if someone can show me mathematically where I'm going wrong because I'm obviously missing a point that others find quite elementary.