UTH question What would you do?

Playing UTH practice $10 + $10 no Trips.

I'm Holding 7d 4s

Board Qc Qd Js 10s Ks

Call or Fold? Why?

Never mind, I'm a dummy.

Quote: 98Clubs

Playing UTH practice $10 + $10 no Trips.

I'm Holding 7d 4s

Board Qc Qd Js 10s Ks

Call or Fold? Why?

My Decision

I chose to call, and was warned "Incorrect". I ignored, and got a push. This hand just looked funny, Villain with ANY 9 or better is a winner, and 8-high or less is a push. Outside chance holding two spades or pocket pair. I rationalized (perhaps incorrectly) that I should try for the push rather than a guaranteed loss of $10+$10.

Okay, so you have a guaranteed loss of $20 if you fold.

The calculator for UTH available at WoO puts the EV of Folding at:

-2.000000

and the EV of Raising at

-2.169697

Okay, so now we have to look at some probabilities:

The first thing that we are going to consider is any nine or better for the dealer. This is going to be based on deck composition and I'm just going to figure out the probability of the dealer NOT having a nine or better and then subtract that from 1 as I find that easier.

There are 45 cards remaining and in a full 52 card deck there would be 28 cards with a Value of Eight or lower. In this case, there are 26 such cards remaining as you hold two of them. The probability of the dealer not having at least a nine is:

26/45 * 25/44 = 0.3282828282828283, so the probability of the dealer having it is 1 - 0.3282828282828283 or 0.6717171717171717

Okay, so if you call and risk three units, then your EV just based on that .6717171717171717 * -3 = -2.0151515151515

Conclusion

Straights are irrelevant as requiring a 9 or an A.

The determination of Inside Pairs or Two Spades (discluding any combination with As, Qs, or 9s) is irrelevant because your EV is already worse by calling just based on the probability that the, "Villain," has at least a 9. Just for s**** and giggles, though:

88, 66, 55, 33, 22 share probabilities because you hold none of those cards. We determine the probability of one and multiply by 5.

(4/45 * 3/44) * 5 = 0.030303030303030304

We take that result * -3 and get = -0.09090909090909091 units.

We now find ourselves at -2.106060606060591 units.

77, 44 share a probability as you hold one of each. We will determine the probability and multiply by 2.

(3/45 * 2/44) * 2 = 0.006060606060606061

We take that result * -3 and get = 0.01818181818181818 units.

We now find ourselves at -2.124242424242409 units.

Finally, we have to look at the Spade Flush. This is a combination of seven specific cards that does not involve a nine or better, except you have the four, so make it six cards.

6/45 * 5/44 = 0.01515151515151515

We take that result * -3 and get = -0.04545454545454545 units

We now find ourselves at -2.1696969696969543

Which matches the UTH Calculator from WoO.

Fold. While it is a a good board to play (paired board with a high-ish kicker, 4th nut), you also have to account for the eight straight outs the dealer has (4 9s, 4 As).

Sorry for the very late response. I've been working independently on a UTH strategy. When this hand occurred the game advisor said Fold. When I went to use the JS strategy advisor, the popup window doesn't open for me. Not sure why. Coupled with discount's cryptic rulings using incorrect strategy safety nets, I thought to make something more coherent.

Bottom line: this particular example winds up being a bad example for a general case rule. Although the board is a pair with 10-low kicker, the str. and flush probabilities make a boarderline case worse. What is apparently better is a one pair board in which your pocket card makes a 10-low kicker... thus one can actually win a portion of the time. For example the exchange of one card in the example the 10 for my 4, turns this bet around quite alot. So, as I have learned through practice, and here, the BOARD 10-low is a worse bet than a pocket 10-low. This type of ruling becomes quite evident in other aspects of this game. THanks much for the detailed response, I am on the correct path, just needing to fleshout more probabilities.