UTH Strategy

I've been using the Wizards strategy for playing UTH and enjoy the game very much. In trying to understand the rule of 21 for the 1x decision point, I created the following spreadsheet:

--- --------------------------------------------Chance of dealer beating you-----------------------------------------------------------
----------------------------- First chance-------First chance----------Second chance-------- Second chance
Outs---------Cards-------- of catching-------- of not catching------ of catching----------- if first missed--------- Overall
15------------ 45------------ 33.33%------------ 66.67%--------------- 40.91%--------------- 27.27%--------------- 60.61%
16------------ 45------------ 35.56%------------ 64.44%--------------- 43.18%--------------- 27.83%--------------- 63.38%
17------------ 45------------ 37.78%------------ 62.22%--------------- 45.45%--------------- 28.28%--------------- 66.06%
18------------ 45------------ 40.00%------------ 60.00%--------------- 47.73%--------------- 28.64%--------------- 68.64%
19------------ 45------------ 42.22%------------ 57.78%--------------- 50.00%--------------- 28.89%--------------- 71.11%
20------------ 45------------ 44.44%------------ 55.56%--------------- 52.27%--------------- 29.04%--------------- 73.48%
21------------ 45------------ 46.67%------------ 53.33%--------------- 54.55%--------------- 29.09%--------------- 75.76%
22------------ 45------------ 48.89%------------ 51.11%--------------- 56.82%--------------- 29.04%--------------- 77.93%
23------------ 45------------ 51.11%------------ 48.89%--------------- 59.09%--------------- 28.89%--------------- 80.00%
24------------ 45------------ 53.33%------------ 46.67%--------------- 61.36%--------------- 28.64%--------------- 81.97%
25------------ 45------------ 55.56%------------ 44.44%--------------- 63.64%--------------- 28.28%--------------- 83.84%

Dealer beating you on his first card is simply the number of outs divided by the cards remaining (always 45). Assuming he didn't beat you on the first card, beating you on his second card is the number of outs plus additional 3 for matching his first card divided by 44 (45 minus dealers first card). You can't simply add the two together since some of the time he will beat you with either card. Multiplying the second chance times the times he did not match on the first gives the added chances he will beat you.

Without a pair on the board you only have to win 1 out of 4 hands (lose 75% of the time) to break even. This is in agreement with the Wizards strategy. Dealer will win only 73.48% of the time with 20 outs.

But with a pair on the board you only have to win 1 out of 5 hands (lose 80% of the time) to break even since you will get paid for the ante bet. If my calculations are correct you should still bet with 21 or 22 outs. Even 23 outs is an even bet.

I am not a math expert and would appreciate any corrections and or comments.

Please excuse the formatting, it was difficult to get the columns lined up.

Your hand-made chart illustrates correct thinking, but you may not know why. Much like surrender in Blackjack, one needs a 75% loss or greater to surrender. 21 outs is just over 75% and therefore a fold, as opposed to a risk of one additional unit. 20 outs or less is less than 75%, indicating a 1x Raise (a Call).

There are times when you should fold a pair of 2's, 3's, or 4's. For example theres a 4-flush on the board and your pair is the bottom rank. Open-ended Straights can raise the 4's, but if BOTH are on the board (5678K with 578K suited) even 5's will fold if bottom paired. My rather limited (by statistical comparison) use of the UTH Calculator at WoO informs me this is so.

In the case of a 4 flush board, isn't it just a matter of adding the 9 additional outs and fold if total is above 20 or above 22 if there is also a pair on the board. Same for straight, 4 additional for inside straight, 8 for open ended straight.