Full house odds in O8 (cross posted from Probability)

[Billy Baroo]

I posted this in the probability forum, but my question isn't getting any action. Can anyone here help me out? TIA

What's the probability that a set on the flop will make either a boat or quads in 08? For bonus points, if there are two low cards out when you flop top set, what are the chances that no low will be possible if you fill up.

TIA

[Slim Pickens]

I'll try. No promises as to correctness.

[ QUOTE ]
What's the probability that a set on the flop will make either a boat or quads in 08?

[/ QUOTE ]
Assuming you don't already hold any of the potential board pair cards, you have 7 outs.

full house (read or better) on turn: 7/45=15.6%
full house on river given none on turn 10/44=22.7%

turn or river: 0.156+(1-0.156)*0.227 = 34.8%


[ QUOTE ]
For bonus points, if there are two low cards out when you flop top set, what are the chances that no low will be possible if you fill up.

[/ QUOTE ]
Assuming you hold 4 high cards...
given full house on turn, low appears on river 24/44=54.4%
given full house on river, low appears on turn 24/45=53.3%

1-[(0.156*0.545+0.533*0.227)/(0.156+0.227)]=46.2%


Slim

[Buzz]

[ QUOTE ]
What's the probability that a set on the flop will make either a boat or quads in 08?

[/ QUOTE ]

Billy - Basically you need the board to pair on the turn or river.

There are three cases.
(1) You can flop a set with no other cards in your hand of the same rank as the two other board cards.
(2) You can flop a set with one cards in your hand the same rank as another board card.
(3) You can flop a set with two cards in your hand the same ranks as the two other board cards.

Case one: 341/990 is probability of improving to quads or boat.
Case two: 306/990 is probability of improving to quads or boat.
Case three: 270/990 is probability of improving to quads or boat.

Do you want odds?
Case one: 649/341 = ~1.9 to 1 are odds against improving to quads or boat.
Case two: 684/306 = ~2.2 to 1 are odds against improving to quads or boat.
Case three: 720/270 = ~2.7 to 1 are odds against improving to quads or boat.

Unless you’re going all in on the second betting round, usually you should want odds of making your hand on the turn, and then, assuming you missed your hand on the turn, in a separate transaction, you’d want odds of making your hand on the river. But that’s not what you asked for. Whatever.

I don’t need any extra credit. Maybe someone else will do that part for you.

Buzz

[Slim Pickens]

[ QUOTE ]

For bonus points, if there are two low cards out when you flop top set, what are the chances that no low will be possible if you fill up.

[/ QUOTE ]


OK, so two low cards on the flop assumes your set is a high, and the two low cards are of different ranks. If you hold 4 high cards...
given improvement on turn, low appears on river 24/44=54.4%
given improvement on river, low appears on turn 24/45=53.3%

Using the conditional probabilities from the case where you've paired no other board cards...
1-[(0.156*0.545+0.533*0.227)/(0.156+0.227)]=46.2%

As odds that's 1.16:1 for there being no low possible when you improve. If you so have board cards paired, it doesn't change this significantly, but your odds of improving get worse as others have shown.

Slim