WHY?? - Top Pair VS Straight Flush Draw

[IRV]

Can someone expain why this is the case? 7c 8c has 8 outs (6's & J's) plus 7 (9 clubs - 2) = 15 outs. Why is this hand favored over 55%? 15/47. Isn't he a 2.5 to 1 dog??

./pokenum -h Ah Kd - 7c 8c -- 9c tc Ad
Holdem Hi: 990 enumerated boards containing Tc 9c Ad
cards win %win lose %lose tie %tie EV
Kd Ah 442 44.65 548 55.35 0 0.00 0.446
8c 7c 548 55.35 442 44.65 0 0.00 0.554

[MarkGritter]

[EDIT: you fixed the out count.] 6 straight outs and 9 flush outs (13-4) = 15/45. (This situation has 52-7=45 unknown cards, not 47.)

But the draw has two chances to hit. If the draw misses on the turn, the river is still to come. So the probability is about 15/45 + 30/45*15/44 = 0.561.

The true number calculated is a little lower than that because of the cases where AK can redraw to a full house or quads after the Ac or Kc.

[BruceZ]

[ QUOTE ]
Can someone expain why this is the case? 7c 8c has 8 outs (6's & J's) plus 7 (9 clubs - 2) = 15 outs. Why is this hand favored over 55%? 15/47. Isn't he a 2.5 to 1 dog??

./pokenum -h Ah Kd - 7c 8c -- 9c tc Ad
Holdem Hi: 990 enumerated boards containing Tc 9c Ad
cards win %win lose %lose tie %tie EV
Kd Ah 442 44.65 548 55.35 0 0.00 0.446
8c 7c 548 55.35 442 44.65 0 0.00 0.554

[/ QUOTE ]

1. There are 45 cards left, not 47.
2. 15/45 is a 2:1 dog to hit the turn, but he has two draws.

Chance of not catching an out is (30/45)*(29/44) = 43.9%, so chance of catching an out is 56.1%. He can also win by catching running 7's, 8's, or 78. This has probability of (6/45)*(5/44). But he loses if he catches exactly JQ with no clubs, and this probability is (6/45)*(3/44). He also loses if his opponent makes a full house with Ac(9,T,K) or AsKc or KcKs or KcKh or quads with AcAs. There are 9+1+1+1+1 = 13 ways to make these hands out of 45*44/2 = 990 draws, so all together we have:

1 - (30/45)*(29/44) + (6/45)*(5/44) - (6/45)*(3/44) – 13/990 = 55.4%.

This works out to exactly 548 winning boards, in perfect agreement with the software.

[IRV]

Thanks to both of you for the explaination.

I guess you really can't do this in your head while playin but rather just memorizing common scenarios?

[Yads]

[ QUOTE ]
Thanks to both of you for the explaination.

I guess you really can't do this in your head while playin but rather just memorizing common scenarios?

[/ QUOTE ]

Yes, just remember the following:
9 outs: just over 1/3 of the time by the river
14 outs: just over 1/2 of the time by the river
21 outs: just over 2/3 of the time by the river

[IRV]

Thanks!