Quick Quads 9/6 DDB Question

[damule]

I was reviewing Quick Quads 9/6 double double bonus strategy on the wizardofodds.com. Can someone please explain to me why holding 4-4-2 is not listed as an equal hold to 2-2-A. In fact, 4-4-2 is not listed as a correct hold at all. It is recommended just to hold 4-4 and drop the 2. I completely understand the strategy of why holding 4-4-A, 4-4-3, 3-3-A, and 3-3-2 have higher value than 2-2-A. I'm completely puzzled as to why 4-4-2 is not listed equally to 2-2-A and not at all. 
 
Any help with this quandry would be appreciated. 

[BillyJoe]

Just a guess, but your 4-4 could QQ with a 3-A, where you do not need a 2. For 2-2, only an A would complete a QQ.

[Eduardo]

Good job billyjoe. I was stuck thinking about this one for a while. I knew the 2 was like a "penalty card" for a quick quad with the 4's, but so is the Ace to the 2's. I didn't stop to think about the other QQ combos.

[New2vp]

Damule, You are absolutely correct that 44+2 and 22+A are equivalent holds.One can infer from bj's post the idea that the value of holding a pair of 4's is not the same as holding a pair of 2's.For 22+A to show up in a strategy chart, its value must beat a the value of a pair of 2's.But even if 44+2 is better than a pair of 2's, for 44+2 to show up in the strategy chart, it must be better than a pair of 4's.In this game, the value of holding 44+2 is higher than the value of a pair of 2's but is never the highest hold in any given hand because the value of holding a pair of 4's with a 2 penalty card is still higher than the value of holding 44+2.A 2 hurts the value of 44 by the same amount that an A hurts the value of 22.See example holds below for given 5-card dealt hands, followed by expected values of the holds in question when playing 6 coins in 9-6 DDB Quick Quads 44 69T:  44 = 5.601944 692:  44 = 5.4335; 44+2 = 5.2960  44 682:  44 = 5.3938; 44+2 = 5.2960  Conclusion 44 > 44 + 2; never hold 44 + 222 69T: 22 = 4.876722 69A: 22 = 4.7083; 22+A = 5.2960  Conclusion 22 + A > 22An interesting situation in TDB Quick Quads (both 9-6 & 9-5) is that sometimes you hold 22+4.  This won't happen if there is an Ace in the hand, since 22+A is better than 22+4, but when a 3 is present in the hand, that weakens the 22 hold enough to make 22+4 better than 22.  And in the same hand 22+4 > 22 + 3 because of the 359 to 1 shot at drawing two more 4's and getting 22444, Quick Quad 4's, whereas drawing two more treys to 22+3 yielding 22333 would merely be a full house.

[damule]

Thanks New,
 
As always you back up the data with sound mathematical evidence. But, for some reason, it still appears to my naked eye that the 4-4-2 hold should equal the 2-2-A hold.
 
billyjoe had a good point where the 4-4 hold alone has more quick quad opportunities left in the draw with four aces and four threes still rolling around (hence 8 cards); as well whereas with 2-2, the only quick quad opportunities in the draw are the remaining aces.
 
I think I can finally see the light.
 
Also New, thanks for the special plays you mentioned with TDB, as this is a game I do play around with abit as well.

[New2vp]

Thanks New,  As always you back up the data with sound mathematical evidence. But, for some reason, it still appears to my naked eye that the 4-4-2 hold should equal the 2-2-A hold.Mule, you're welcome.  Neither of us posts as often as we used to; I might not have posted here if you weren't the one to ask the question.  And with my current work load, I think I will once again have to go into "forum hibernation," so this might be my last post for a while.442 DOES equal 22A.  That portion of the strategy chart could look like:1.  442.  333.  442, 22A4.  22It's just that since 44 is above 442, 442 never gets to play.  So an equivalent strategy chart would be:1.  442.  333.  22A4.  22And it could be abbreviated further to be:1.  22A2.  22-44This last chart would then cause confusion because it appears that 22A is above 44 when it really isn't.  22A simply is never compared to 44 (or to 33 since 332A is a more valuable hold).  If 44 and 22A were in the same hand, you would hold 4422, Quick Trip 4's.