Well, my quiz (as well as Rhound's original hand, which is actually more complicated since 3-way) bring up yet another question, which I'd like to examine in a HU context: Assuming that you have Pt as tie probability, what do you need as win probability (Pw) to call a pot-sized bet, assuming that no further betting is possible?

Basically, the pot is $100, you have $50, and it costs you $50 to call with the given winning and splitting probabilities. All others have folded.

Ok, in case of a split, you get back $75. So, you're actually making a little money on the split--namely, Pt*25.

In case of a win, you get the full $100 in the pot, so you make Pw*100.

And in case of a loss, you lose (1-Pt-Pw)*50.

Total EV is thus Pw*100 + Pt*25 -(1-Pt-Pw)*50 = Pw*100 + Pw*50 +Pt*25 + Pt*50 - 50 = Pw*150 + Pt*75 -50.

For the break-even point, we thus get:

Pw*150 + Pt*75 - 50 > 0 iff
Pw*150 +Pt*75 > 50 iff
Pw*3 + Pt*(3/2) > 1

So, for example, if the probability of a split is 50%, we have Pt*(3/2) = 75%. Hence, Pw*3 must be greater than 25%. So, you need a win probability of 9% or more to break even.

my head hurts after reading this.