My thoughts on hand 1:

If you have 10 outs to improve to boat or quads with one to come, that's 10/46 or a 21.7% chance of improving -- or a little better than 4 to 1 against -- not 17%. The pot is laying you 47 to 20, or 2.35 to 1, so it's obvious that if you have to improve on the river to win, folding is obviously correct. However, the situation isn't so simple. Let's say 20% of the time your opponent is bluffing here and only has 2 pair or worse and the other 80% he has the straight (for the sake of simplicity, we will not consider the possibility of a higher set). Let's also say your opponent will at least pay you off for another 20$ on the river if he's got the straight and the board pairs.

If he is bluffing and you call:

78% of the time you won't improve. However, there are 9 clubs that could fall on the river which would stop all but the most aggro opponents from betting again on the river if checked to unless he was semi-bluffing with the flush on the turn. So, let's say 1/2 of the time your opponent was bluffing with the 4 flush. 10% of the time, then, the flush will hit him and he will bet and you can safely fold. 10% of the time the flush will miss him and he will (unless insane) check behind to your river check and you will win a big pot. The last consideration is how often your opponent will fire twice into this pot on a bluff or with a hand weaker than yours when a non-club comes on the river (80% of the time), and whether you will call this second bet. Let's say 50% of the time he will fire again on the river with a hand weaker than your set, and 50% he will check behind if checked to. Let's say he will bet 20$ when he does choose to fire again and that you'll call the additional 20$. Therefore, when he is bluffing and you miss and no club falls, 1/2 the time you will be facing 20 and 1/2 the time he will check behind. So, 40% of the time when he was bluffing the turn you will win an additional 20$ (or 87$ total), 50% you will win 67$, 10% you lose 29$. Therefore, the EV of calling when he is bluffing and you miss is 34.8$ + 33.5$ - 2.9$ or 65.4$.
For the sake of simplicity, let's say if the board pairs on the river he will not pay off any more, meaning you win 67$. Therefore, total EV for bluffing is (.2)(.22)(67) + (.2)(.78)(65.4) = 2.948 + 10.2024 = 13.15 or so.

If he is not bluffing and he does have the straight and you call:

22% of the time the board pairs. When this happens we already established he will call an additional 20 for a total winning of 87$.

78% of the time the board doesn't. However, like above, if he has the straight he almost certainly doesn't also have a flush draw and he might very well put you on that. So, if a club comes, which will happen 20% of the time and the board doesn't pair and you check to him he will almost certainly check behind, meaning you only had pay 20 instead of 40 to see the river (as we already established you would call another 20 above). However, if the board doesn't pair and he is not bluffing and a club doesn't come, your opponent will certain make another bet. Let's say 1/2 of the time this is just another 20, and half of the time it's allin, or about 40. To make the analysis simpler and more realistic, let's assume that if he dumps allin on the river he can beat your set, so you always fold, but you always call the 20. So, if the board doesn't pair and he is not bluffing: (.8)(.78)(.2)(-29) (club hits and he checks behind) + (.8)(.78)(.8)(.5)(-49) (no club and he bets 20) + (.8)(.78)(.8)(.5)(-29) (no club and he bets 40) = -3.62 -12.2304 - 7.2384 = -23.088.

Finally, if you hit and he has the straight, we said he will call another 20: (.8)(.22)(87) = 15.312.

Total EV for call is around 13.15 + 15.312 - 23.088 = $5.374.


If all of my assumptions are valid (which they aren't), calling here seems marginally correct, winning you around 5.37 dollars on average. That said, if you think he is bluffing less than 20% of the time (which might be very valid), or there is a chance he's got AA in which case you're drawing almost dead, then it appears calling or folding are almost even EV and you aren't losing any money by folding as you did. Conversely, if he is a guy who makes big bluffs, overvalues 2 pair or top pair, or thinks you're a brick, calling here is even more correct.

I'm curious to hear what the mathematicians on this site think of this analysis.