? About Shoving and FE

Alright I was going to hijack Razvan's thread about the nit turn raise with his set, but figured it more appropriate to just start a new thread. But for the sake of ease I'll be using the numbers from his hand to ask this question since the cards aren't entirely necessary here.

What I'm wondering is that obviously when we are shoving for value we want to beat >50% of villains calling range since we stand to win any money in the pot over half the time if this is the case making us $$$ over time, okay basic.

However, is there a time when we'll shove that might look like a value shove when we have less than 50% equity? Suppose we only have 40% equity vs Villain's calling range can we figure out a BE point of FE we need from our shove to cover the difference?

Using Razvan's example, he got raised on the turn and has $8.17 left behind. The pot at that point is $6.25. So here's my maths for this, just checking to see if this makes sense though...


If on a shove we only beat 40% of villains calling range, we stand to only make 40% of the pot over time so total pot is (6.25 + 8.17 + 8.17) = 22.57 * 40% = $9.04

If we had the 50% we needed to break even it would be 22.57 * 50% = $11.30

The difference between the two being $2.26. So we need to make that amount in FE out of the original pot which was $6.25. So 6.25/2.26 = 36% FE required to break even.

Or do we calculate our FE from the risk 8.17 to win 6.25 idea where we have 8.17/(6.25+8.17) = 56% FE required to make a EV play here with our only have 40% equity vs Villain's calling range? I feel like it's the first idea on FE I presented... but am definitely open to being wrong. This second FE calculation I think assumes we have relatively 0% equity against villains calling range.

So beyond the math here, does this idea even make sense or am I just completely off track here?