Difficult dealt hands

[Eduardo]

[olds442jetaway]

That’s for sure. I think that term no matter where or how the word ugly is used by us guys, it sets off all kinds of possible bad moods and even the more deadly…..yes and you did this or that….even if it was 37 years ago. Well, now since I have just shot myself in the foot with all of the beautiful ladies who post here, I guess I better retire into me dog house for the evening.

[asteroid]

The quick answer is that you (or, more realistically, some software) need(s) to evaluate both (1) the expected value of holding 98 suited and (2) the expected value of tossing all 5 cards and drawing a new hand for all possible card combinations that do not have holds with even higher expectation; and then choosing the hold that has maximum value. It is interesting that your username is "details" as most people would give up trying to deal with (2) above; however, there generally is enough information in the Pro Training option that you used to pose your question if you have the patience and develop the know-how to deal with it.

I am going to guess that Bob Dancer may have answered this or a similar question some time over the years, but following is what I put together a few years ago to determine whether to hold 98 suited or draw 5 new cards:

Divide the ranks up into these sets:

A = {6, 7, T, J}
B = {5, Q}
C = {3, 4, K, Ace}

You can see that the deuce is not in any of those sets; if there is a deuce present, of course, you would hold it along with 98 unless you had an even better hold available.

If any of the other 3 cards were suited with the 98, you would either have a 3-card straight flush draw to hold (if the rank of the suited card were in sets A or B) OR you should toss all 5 cards (if the suited card was in set C) since the presence of that suited card would reduce the value of holding the 98 suited sufficiently to prevent it from being the better hold.

Then there are 3 relatively simple rules to follow (you can look above in the list that dinghy provided for comparison):

1. If any of the 3 other cards are in set A, toss all 5 cards.
2. If the 5 and Q (from set B) are BOTH included in the 3 other cards, toss all 5 cards.
3. If all the other 3 cards are in set C, keep the 98 suited.

(A perhaps obvious exception to Rule 3 would be if the other 3 cards were A34 suited, because holding A34 would have a higher expected value than the 98.)

Those 3 rules cover almost all situations. The only ones left are situations in which one of the cards is either the 5 or Q and the other 2 are in set C. An easy answer would be to toss all five cards in these situations also, but that would only be right 65% of the time with a 5 and 56% of the time with a Queen.

Below are the more detailed rules for when you would hold the 98 in those situations. Here you also have to note that there are better holds available once in a while. With the 5, a suited 543, 5A3, or 5A4 is better than 98 suited. And with Q, you have to pay attention to holding KQ suited sometimes.

What remains are these more complicated rules.

With a 5:

If the other 2 cards are:
1. K4, keep the 98 suited UNLESS 5K4 are all of the same suit.
2. AK, keep the 98 suited UNLESS the A and K are of the same suit.
3. K3 or A4, ONLY HOLD 98 suited if the 3 discards are of all different suits.
4. 43 or A3, draw 5 new cards.

With a Q:

If the other 2 cards are:
1. K4, keep the 98 suited UNLESS you can hold KQ suited.
2. AK, keep the 98 suited UNLESS the A and K are of the same suit.
3. 43, A3, K3 or A4, ONLY HOLD 98 suited if the 3 discards are of all different suits.

Holding the 98 suited with either a 5 or Q and 2 cards from set C has the same expected value regardless of the ranks and suits of the other 3 cards.

So, these more complicated rules have to do with the expected value of drawing 5 new cards from the 47 remaining cards after the deal. That expected value is lower when there are more suits in the 5 dealt cards and when there are more cards in the middle ranks than the end ranks. These principles can help in understanding the similar analyses that would need to be completed for similar potential holds comparing T9 suited, 87 suited and 76 suited to tossing all 5 cards. A warning: the rules for these will be similar but slightly different than what I've shown for 98.

Clearly, there is a lot of work that would still need to be done. And many could logically determine that it is not worth the trouble to them for their purposes. However, if you want to try for 100% accuracy, there is a requirement of understanding and following these additional rules. Plus, you may have to figure out how to do such comparisons. Best of luck.

Great post New2vp. Here's a combinatorial addendum to your post (possibly redundant):

[New2vp]

Thanks asteroid, I don't think your addition is redundant at all. It was through software like this that I was able to teach myself many years ago how the various 3-card combinations of discards affected the 2nd line above that you have outlined.

Seeing exactly how tossing all 5 cards, indicated in your illustration by "1.6053 _____", showed different numbers of flushes, straights, straight flushes, and royal flushes (wild or natural), with the differences dependent on the ranks and suits of the 3 discards, was the major key in understanding how the expected value of the hold changed to determine optimal holds for these difficult-to-play dealt hands.

[asteroid]

Thanks asteroid, I don't think your addition is redundant at all. It was through software like this that I was able to teach myself many years ago how the various 3-card combinations of discards affected the 2nd line above that you have outlined.

Seeing exactly how tossing all 5 cards, indicated in your illustration by "1.6053 _____", showed different numbers of flushes, straights, straight flushes, and royal flushes (wild or natural), with the differences dependent on the ranks and suits of the 3 discards, was the major key in understanding how the expected value of the hold changed to determine optimal holds for these difficult-to-play dealt hands.

Cheers m8.