Adams Myth wrote:
I was looking up Double Joker Poker on WinPoke.
Came across this hand. Nothing special about it.
Joker-K4 of Clubs, 9 of Diamonds, 7 of Spades
In the Details window, it shows that W9 could result in 5996 3-of-a-
Kinds, and 1086 Straights, whereas the W7 would have 6000 and 1050
respectively. All the other paying hands are the same for either
hold, namely 252 Flushes, 189 Full Houses, 372 4-of-a-Kinds, 34
Straight Flushes and 4 5-of-a-Kinds.
I can understand the difference in Straights, though I haven't
worked out the numbers (never mind WinPoker has already done that!),
but the number of 3-of-a-kinds is, I mean the difference,
surprising. One would think holding a W9 or W8, or for that matter a
Wx should not change the number of 3-of-a-kinds it produces.
What's the intuitive explanation?
I changed the 7 to an 8. Now the hands are equal in all respects.
They both produce the same number of 3-of-a-kinds, 6000.
Of course I am not questioning the software. I am just looking for a
simple, intuitive explanation.
I expect after a little more contemplation you'll have no difficulty
working this one out -- and I'm tempted to suggest you work on it a
little more since I think you'd find solving it rewarding.
However, expecting that someone else might otherwise steal my thunder
here, I'll spiel it out 
Winpoker isn't, of course, giving you a count of the total number of
Straights or 3K's that can be formed on the draw. It's giving you the
number of WINNING Straights/3K's. In those cases where a final hand
should yield both (which would involve the draw of the 2nd Joker), the
Straight will be the better of the two and would be the only one
that's counted.
As you say, in any case, the number of 3K's that can be formed on the
draw is identical, no matter what card we're talking about. However,
when we're talking about a case where a greater number of straights
can be formed in general, the number of 3K's that will come out as
winners will be smaller.
As you saw (and rationalized) a greater number of straights are
possible when holding W-9 than holding W-7. This is because in either
case the 4c has been discarded and isn't available on the draw. That
reduces the possible W-7 straights, but not the W-9. Since there are
a greater number of W-9 straights, this reduces the number of winning
3K's when those are the held cards.
I'll leave the exercise to you of determing why, when there are 36
more winning straights with the W-9, there are 4 fewer winning 3K's.
Another head slapper for you, should it not be apparent right off the
bat, is that under the same logic it might actually be surprising that
both the W-9 and W-8 holds yield the same number of winning 3K's, as
you went on to note (all other cards held constant).
In tossing the 4c, it's still the case with W-8 that tossing the 4c
reduces the number of possible straights but, as discussed, doesn't
impact the W-9 straight possiblities. So, shouldn't it follow that
this would result in more winning W-9 straights and thus fewer winning
3K's for the W-9 hold?
(Hint: Extend the logic presented above.)
- Harry