[switters]

so, let's say you have T[img]/forums/images/icons/heart.gif[/img] J[img]/forums/images/icons/heart.gif[/img] :

# of remaining cards : 50
# of remaining [img]/forums/images/icons/heart.gif[/img]s : 11

number of flops containing precisely two [img]/forums/images/icons/heart.gif[/img]s:
(11 choose 2) * 39 = 11 * 10 * 39

number of possible flops:
(50 choose 3) = 50 * 49 * 48

percent of flops containing precisely two [img]/forums/images/icons/heart.gif[/img]s if you hold two [img]/forums/images/icons/heart.gif[/img]s :
(11 * 10 * 39) / (50 * 49 * 48) = 3.64%

FYI, the percent of flops containing AT LEAST two [img]/forums/images/icons/heart.gif[/img]s if you hold two [img]/forums/images/icons/heart.gif[/img]s (this counts the times you flop a flush)
(11 * 10 * 48) / (50 * 49 * 48) = 4.49%

now, we need to count the open-enders you will flop. take for example, 89 (but not 789 or 89Q, since those aren't open-enders, they're straights...)

number of 8's : 4
number of 9's : 4

number of open-enders with 89: 4 * 4 * 40 (40 = 48 remaining cards minus the 4 7's and 4 Q's)

the same logic applies to the other two ways to flop open-ended:

number of open-enders with Q9: 4 * 4 * 40 (40 = 48 remaining cards minus the 4 8's and 4 K's)

number of open-enders with KQ: 4 * 4 * 40 (40 = 48 remaining cards minus the 4 9's and 4 A's)

percentage of open-ender flops:
3 ( 4 * 4 * 40) / (50 * 49 * 48) = 1.63%

percentage of flopped straights (four ways, 789, 89Q, 9QK, QKA):

4 ( 4 * 4 * 4 ) / (50 * 49 * 48) = 0.22%

now, we can't just sum up the percentages of flopped 4 flushes and flopped 4 straights, because there is overlap between those possibilities...

for 89, for example, we must subtract out:
8[img]/forums/images/icons/heart.gif[/img]9[img]/forums/images/icons/heart.gif[/img]Xo = 1 * 1 * 33
8[img]/forums/images/icons/heart.gif[/img]9oX[img]/forums/images/icons/heart.gif[/img] = 1 * 3 * 8
8o9[img]/forums/images/icons/heart.gif[/img]X[img]/forums/images/icons/heart.gif[/img] = 3 * 1 * 8

and, if we are counting flopped flushes, we must subtract out the four-straights that are flushes...

8[img]/forums/images/icons/heart.gif[/img]9[img]/forums/images/icons/heart.gif[/img]X[img]/forums/images/icons/heart.gif[/img] = 1 * 1 * 7


and for the 256 times you flop a straight, we need to subtract out the 4 straight flushes [img]/forums/images/icons/wink.gif[/img] and the
12 times you flop a straight and a four-flush...

so, the grand total is...

percent of time you will flop precicely a four-flush or four-straight:
[ (11 * 10 * 39) + (3 * 4 * 4 * 40) - (3 * (33 + 24 + 24)) ] / (50 * 49 * 48) =
[ 4290 + 1920 - 243 ] / 117600 = 5967/117600 = 5.07%

percent of time you will flop AT LEAST a four-flush or four straight (inludes flopping straights and flushes, but doesn't count flopping trips, boats, or quads, for example):
[ (11 * 10 * 48) + (3 * 4 * 4 * 40) + (4 * 4 * 4 * 4) - (3 * (33 + 24 + 24 + 7)) - 4 - 12) ] / (50 * 49 * 48)
[ 5280 + 1920 + 256 - 264 - 4 - 12 ] / 117600 = 7176/117600 = 6.10%

it's quite possible that, in my rush, i've made a small error in the double-counting subtractions, but this is basically the right process for answering probability questions like this, and the actual numbers I calculated should be very close to correct, probably within 0.25%...

-switters