Flush vs a Full House or Quads draw

DistantHaze · #1 02-10-2006, 11:21 AM

I have a question for any math wizards in here: what is the total % chance of a pocket pair beating a flopped flush? For example, if you hold A6s, I hold KK, and the flop is 4s Ks 3s, the odds of a Full House or Quads change depending on the Turn card. For the turn card to pair the board, there are 7/45 cards (3x4, 2xK, 3x3). If the Turn is a 4, for example, then the chance of catching a Boat on the River is 4/44 (2x4, 2xK). If the Turn does NOT pair the board, however, and is an 8 for instance, then the River will make a Full House or Quads 10/44 times (3x3, 3x4, 3x8, 2xK). How do you calculate the COMBINED chance of the pocket Kings beating the Flush?

Tom1975 · #2 02-10-2006, 01:58 PM

7/45 + (1-7/45)*10/44=34.74%

DistantHaze · #3 02-14-2006, 06:09 PM

Thank you, Tom. Can you explain how you came to that answer?

Tom1975 · #4 02-14-2006, 06:14 PM

On the turn you 7 out of 45 cards which improve your hand to a boat/quads. If you do you hit on the turn, you no longer need to worry about what happens on the river. You only need to worry about the river the 1 - 7/45 times you don't hit the turn. In this case there are 10 out of 44 cards which will improve your hand. So the total odds are 7/45 + (1-7/45)*10/44

Pappy25 · #5 02-15-2006, 01:04 AM

Odds of making your boat or quads on the turn would be 5.7 to 1 having 7 outs out of 47 cards not 45, or 14.9%. If you know he has the flush then you are dealing with 7 out of 45 which increases your percentage to 15.5%. Making the boat or quads on the river is 3.6 to 1 or with you knowing the flush is there the 10 out of 44 which is 22.7%. To get the combined percentage of you winning the hand after the flop just add them up. So you total chance of improving to a boat or quads after the flop when up against a known flush would be 38.2%.

DistantHaze · #6 02-15-2006, 07:12 PM

Perfectly logical. Thank you all for your replies. As an aside, my friend's KK lost to the idiot who called an all-in re-raise with A8s. Flop was Ks 5s 9s.

AaronBrown · #7 02-16-2006, 08:56 PM

KK will win 68% of the time against A8s. Once that flop came down, his chances were cut in half, to 34%.