easy one - prob of trips from one card

[memphis57]

If I'm dealt one ace in the hole, what is the prob I will make trip or quad aces on the flop? I calc 0.002449 by 3*2*48/(50*49*48). Is that right?

Then, if I miss on the flop, what odds by turn and river?

[BruceZ]

[ QUOTE ]
If I'm dealt one ace in the hole, what is the prob I will make trip or quad aces on the flop? I calc 0.002449 by 3*2*48/(50*49*48). Is that right?

[/ QUOTE ]

No, it should be:

(3*2*47*3 + 6)/(50*49*48) = 0.007245.

There are 47 non-aces left, not 48. Then you need to multiply by 3 since the non-ace can occur in any of the 3 positions, and since your denominator counts all possible orders of the 3 cards. Then add 6 for the 6 ways to order 3 aces to make quads (3*2*1). I would normally do it this way with combinations:

(3*47 + 1)/C(50,3) = 0.007245

where C(50,3) = 50*49*48/6 is the number of flops ignoring order. Then there are 3 pairs of aces that can combine with 47 non-aces, plus 1 flop that gives quads.


[ QUOTE ]
Then, if I miss on the flop, what odds by turn and river?

[/ QUOTE ]

It depends on whether you got an ace on the flop. If you got no aces on the flop, then it's 3/47 * 2/46 = 0.28% that you get aces on both the turn and the river.

If you got 1 ace on the flop, then it is 2/47 that you hit on the turn. If you miss the turn, then it's 2/46 that you hit on the river. It is 2/47 + (45/47)*(2/46) = 0.84% that you hit on the turn OR miss on the turn and then hit on the river.

[memphis57]

Thanks, Bruce. Here's the specific problem I'm working on (between hands) - Pacific is offereing an $88 bonus if you win a hand with 888 and $888 with quad 8s. If I'm dealt 8-rag, when should I stay in the pot? Ideally I'm looking for a number to add to the pot and then use pot odds to decide.

I'm roughing up some calcs and will post them soon, but I thought I'd give you the whole problem in case you or anybody else wants to work it out while I'm stumbling around on it.

[memphis57]

Damn! Just missed. Held A8o to the turn against the normal pot odds, hit an 8 on the turn, another on the river, but then lost the hand to 4s full of 8s.

BTW, I'm playing .50/1, you'll have to know that for the calcs, so the bonuses are 88 and 888 BBs.

[BruceZ]

[ QUOTE ]
Damn! Just missed. Held A8o to the turn against the normal pot odds, hit an 8 on the turn, another on the river, but then lost the hand to 4s full of 8s.

BTW, I'm playing .50/1, you'll have to know that for the calcs, so the bonuses are 88 and 888 BBs.

[/ QUOTE ]

What happens if you split?

[memphis57]

Don't know. It says "hands won" so I'd assume the worse and say splitting isn't winning, so no bonus.

[memphis57]

Okay, see if I'm on the right path - I calculate I should add 3.28 BB to the pre-flop pot when betting to see the flop.

There are 3 outcomes that add value over the regular value of my hand:

flop comes 888, odds 1/19600 to make $888, thus value of $0.045

flop comes 88x, odds 3*47/19600 or 0.72% to make $88 AND have 1/47+46/47*1/46 chance of another $800 if it hits quads by the river, total value $0.878

flop comes 8xx, odds 3*47*46/19600 or 33.09%, to have 8.4% chance of making trips by the river, total value $3.39

I left out as insignificant the chance of 8xx flop ending in quads, and of xxx flop ending in trips.

Then I have to discount the value for 2 factors. First, my biggest number of $3.39 from flopping 8xx will sometimes be worthless because the betting will be such that I won't continue. I'll put that at 80% and thus value that piece at .8 * 3.39 or 2.71.

Second, even if I hit trip 8s I could lose the hand, let's call that a 10% chance. So 2.71+0.88+0.05 = 3.64 * 90% = 3.28 add to pot.

[memphis57]

Hmmm, come to think of it, add-to-pot is not the right approach. What's really happening is that some of your outs have far greater value than others. But I can't think of an easy way to measure it. Those leading to a win with 3 or 4 of the rank get a bonus of several times the pot, so it's like you have to add that ratio to your outs.

Difficult problem.

[BruceZ]

[ QUOTE ]
flop comes 8xx, odds 3*47*46/19600 or 33.09%, to have 8.4% chance of making trips by the river, total value $3.39

[/ QUOTE ]

This should be (3*47*46/2)/19,600 since now you are ignoring order in the denominator, so you must do the same in the numerator. Then multiply by (2/47 + 45/47 * 2/46)*88, and this part comes to just $1.22, not $3.39.

[ QUOTE ]
Then I have to discount the value for 2 factors. First, my biggest number of $3.39 from flopping 8xx will sometimes be worthless because the betting will be such that I won't continue. I'll put that at 80% and thus value that piece at .8 * 3.39 or 2.71.

Second, even if I hit trip 8s I could lose the hand, let's call that a 10% chance. So 2.71+0.88+0.05 = 3.64 * 90% = 3.28 add to pot.

[/ QUOTE ]

The possibility of losing to or splitting with another 8 is an even larger discounting factor. With 9 opponents, of the times you make trip 8s by the river, 18/45 = 40% of the time the 4th 8 will have been dealt to your opponents. If they play any 8 rag, and you play any 8 rag, then you will lose or split over half of these or >20%, and even though you don't lose money when you split, you do not have the large equity you are assuming from the bonus. Of course this factor will not apply to your good kickers.

[gamble4pro]

For the flop:
favorable combinations for trips or quads:
(AAx) - C(3,2)*46=138
(AAA) - C(3,3)=1
Totally, you have 139 favorable combinations from C(50,3) = 19600 possible. So, the odds are 139/19600=0.70%
If you miss the flop (no ace at all):
favorable combinations for turn+river cards (this means the "or" event) to generate trips (quads not possible):
(AA)- C(3,2)=3 from C(47,2)=1081 possible. This means 3/1081=0.27% odds.