The second edition of Volume 2 covers Double Bonus--including 10/7, 9/7, and 10/7 returning 80 for a straight flush--with standardized notation affording greater clarity and precision. A couple of strategy errors discovered in the first edition have also been corrected.
a. Kh Qh Jh Js 5c --- $4,260 --- here we have a suited KQJ versus a high pair with no interference whatsoever. What does "interference" mean? If the 5c were a heart instead of a club, that would have made getting a 5-card flush more difficult starting from KQJ and that would have changed the BP to $4,530. A 9 penalty ($4,340), a T penalty ($4,420), and an A penalty ($4,355) are all forms of straight penalties and make it difficult to complete a straight. The only other type of penalty when you have KQJ versus a high pair is a double straight penalty --- such as Kh Qh Jh As Ac. The BP for this would be $4,205.
b. Kh Qh Jh Js 5h --- $4,530 --- this is a case of KQJ versus a high pair that was discussed in the previous paragraph.
c. Js Ts 7s 5h 3c --- $8,115 --- this is a case of JT versus JT7 with no interference. Flush interference isn't important in this kind of a hand because adding another spade would give us a 4-card flush which will "always" be held versus a JT. For straight interference we need to consider an 8 penalty ($7,475), a 9 penalty ($7,795), plus several others including an A penalty, a K penalty, a Q penalty, and several "double penalties" including A+8, A+9, K+8, K+9, and Q+8.
Let's say you memorized these numbers. Would they work well for 8/5 Jacks or Better? The answer is no. In every case, the value of the flush was an important part of the calculation. It was hidden in the formula, but WinPoker used that value to calculate the EVs we started with. Since 8/5 Jacks returns 5 for a flush rather than 6, the numbers will be different.
How about 8/6 Jacks or Better then? That is a game where flushes pay the same as in 9/6, but full houses return 8 instead of 9. The answer is still no, because the value of a full house is an important factor in the value of JJ in the first two hands (JJ becomes a full house 165 out of 16,215 times) and the value of JT in the third hand (which becomes a full house 18 out of 16,215 times).
How about 9/6 Double Bonus or 9/6 Double Double Bonus then. Both of those games return the same for full houses and flushes as 9/6 Jacks does. The answer is still no. Four-of-a-kinds return more in those games and two pair returns less. It all matters.
Consider the following hand in dollar 10/7 Double Bonus: Ah Ks Ts 8c 6d. Holding the A by itself is the better play with a royal of $4,000, but the value of the A increases by a factor of 1/178,365 as the royal increases. The value of KT increases by a factor of 1/16,215 as the royal increases --- a factor eleven times as great. The formula we gave last week won't work because that assumed that the value of one of the combinations would stay constant while the value of the royal increased. Here is the new formula:
10 x DRF = 178,365 x D$EV
You might wonder where we came up with the value of "10". Since in every 178,365 units, the value of A changed by 1 and the value of KT changed by 11, the DIFFERENCE between them increased by 10.
At a royal of $4,000, the value of the A is $2.2782 and the value of KT is $2.2270. Since this is a difference of 0.0512, our formula solves to DRF = 913. If we plug a royal of 4,913 into WinPoker and enter the hand, both A and KT show a value of $2.2833. Voila!
As before, a presence of a heart (for a flush penalty), a card between 2-5 (low straight penalty), two low straight penalties, or low straight flush penalty, a low straight penalty PLUS a low straight flush penalty, or a 9 (which is a straight penalty to the KT) all affect the final answer. But the values provided by WinPoker take that all into effect.
For those of you who find today's column too mathematical, you'll be relieved to know that next week's column will have significantly less math in it. I promise.
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