A few probabilities, simulations and other mathematics.

BeerMoney · #1 01-25-2005, 04:11 PM

I want to get a post going where we have useful probability calculations as well as some useful sims. In the sims in the back of 7CSFAP they don't make too many assumptions about dead and live cards.

Some probabilities.
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Let's say you have trips on 4th street, the chances of filling up or better: (ALL LIVE CARDS) Assume you've seen 13 cards.
P(Filling up) = 1 - P(not filling up)
= 1 - (35/39)*(32/38)*(29/37)= .4077
Odds: 1.45:1 against.

This actually seems high. (Can someone verify.)
************************************************** ******
P(Hitting flush with 4 to a flush on fourth street given 2 of your suit are out from a total of 13 seen cards)= 1 - P(Not hitting flush) = 1 - (32/39)*(31/38)*(30/36) = .4421

Odds : 1.26 : 1

P(Hitting full house with two pair on 4th street, using the two pair you have on fourth street.) Assume all pair cards are live:

P(Filling up or better) = 1 - P(Not filling up)
(Assume 12 seen cards)
= 1 - (36/40)*(35/39)*(34/38) = .2773
Odds : 2.6 : 1

** I would say the prob. of filling would be a little higher since you could go running trips, and that isn't accounted for in my calculation.

Maybe add : 3/39*2/38 = .004 to the above prob. (Negligible.)

******************************************

Some sims.

[a [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 2 [img]/images/graemlins/heart.gif[/img] Vs q [img]/images/graemlins/club.gif[/img] q [img]/images/graemlins/diamond.gif[/img] 5 [img]/images/graemlins/heart.gif[/img]] / DEAD CARDS: J [img]/images/graemlins/spade.gif[/img] 4 [img]/images/graemlins/diamond.gif[/img] 6 [img]/images/graemlins/club.gif[/img] 7 [img]/images/graemlins/heart.gif[/img] t [img]/images/graemlins/diamond.gif[/img] 8 [img]/images/graemlins/spade.gif[/img]
7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
As 2d 2h 221268 44.25 278728 55.75 4 0.00 0.443
Qc Qd 5h 278728 55.75 221268 44.25 4 0.00 0.557
************************************************** *****
Same race, except there is a dead ace..

[a [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 2 [img]/images/graemlins/heart.gif[/img] Vs q [img]/images/graemlins/club.gif[/img] q [img]/images/graemlins/diamond.gif[/img] 5 [img]/images/graemlins/heart.gif[/img]] / DEAD CARDS: A [img]/images/graemlins/diamond.gif[/img] 4 [img]/images/graemlins/diamond.gif[/img] 6 [img]/images/graemlins/club.gif[/img] 7 [img]/images/graemlins/heart.gif[/img] t [img]/images/graemlins/diamond.gif[/img] 8 [img]/images/graemlins/spade.gif[/img]

cards win %win lose %lose tie %tie EV
As 2d 2h 198133 39.63 301862 60.37 5 0.00 0.396
Qc Qd 5h 301862 60.37 198133 39.63 5 0.00 0.604

Effective odds:

For 3/6 party: Assume a full table.
$4 ante, assume completed bet, dead $1 bring in, and assume action goes bet-call, bet-call, etc..
$3 on 3rd, fourth st, $6 on 5th, 6th, 7th..

You are paying $24 to win $4+$1+$24 = $29 Effective odds are $29: $24. Which means you would like chances of winning to be > 24/(24+29) = 45%. (Here I am assuming you play about equally as well as your opponent.)

Please point out any flaws, and add some other sims and probs. to this thread.

imported_big ben · #2 01-25-2005, 04:32 PM

[ QUOTE ]
Let's say you have trips on 4th street, the chances of filling up or better: (ALL LIVE CARDS) Assume you've seen 13 cards.
P(Filling up) = 1 - P(not filling up)
= 1 - (35/39)*(32/38)*(29/37)= .4077
Odds: 1.45:1 against.

This actually seems high. (Can someone verify.)

[/ QUOTE ]

I think I've found what you're looking for- the 35/39 will not go to 32/38, but to 32/(39 - # of dealt cards [not one]).

Ashley Adams has this as 1.5:1 on 3rd, and the only prob. you have to remove from 3rd to 4th is quads (boat is obviously not possible). Nice post!

7stud · #3 01-25-2005, 06:55 PM

[ QUOTE ]
Odds: 1.45:1 against. This actually seems high.

[/ QUOTE ]
What do you mean by that? Do you mean the 1.45 in 1.45:1 is too high of a number, and you think it should be lower, i.e. you think you have better odds of filling up, or you think the probability of filling up produced by your calculations is too high, i.e. the 1.45 in 1.45:1 should be a larger number?

[ QUOTE ]
1 - (35/39)*(32/38)*(29/37)

[/ QUOTE ]
Why 32 cards that don't help you on 6th street? Assuming your cards are live, for each of your two kickers, you can catch 3 cards to make a full house, and you can catch one card to make quads--that's a total of 7 cards that make a full house or better. Also, as the last poster recognized, you see at least one more upcard in your opponent's board, so the denominator is smaller than or equal to 37.

In addition, if your trips and kickers are within reach of each other, there are cards on 5th street and potentially the later streets that can help you make a straight flush(of course, you can assume that away).

Likewise, on 7th street, I think there are 10 cards that can make your hand.

BeerMoney · #4 01-25-2005, 07:31 PM

7 stud..

[ QUOTE ]
Why 32 cards that don't help you on 6th street? Assuming your cards are live, for each of your two kickers, you can catch 3 cards to make a full house, and you can catch one card to make quads--that's a total of 7 cards that make a full house or better.

[/ QUOTE ]

Yes, you're right, I don't know what I was thinking, I was decreasing the numerators by three instead of four..

So, I believe the correct prob is:

1 - (35/39)*(31/38)*(27/37) =.46575

Or, odds : 1.14:1

I hope that is correct.
[ QUOTE ]
Also, as the last poster recognized, you see at least one more upcard in your opponent's board, so the denominator is smaller than or equal to 37.

[/ QUOTE ]

I don't know what you mean by that... You don't know what your opponents upboard will be any more than you know what yours will be. We are talking about the chances of filling up. The fact that you have opponents has nothing to with it, other than the fact that you've seen some of their cards that are not yours. What happens in the future is unknown, and is why we're finding probabilities.

7stud · #5 01-25-2005, 08:26 PM

[ QUOTE ]
Yes, you're right, I don't know what I was thinking, I was decreasing the numerators by three instead of four..

[/ QUOTE ]
It gave me pause for a moment when I considered why your method didn't produce the right result: after all 3 more cards will make your hand on the next street. But of course, it's the amount of bad cards relative to the total cards left that is important, and there is one less card in the deck on each remaining street, which you didn't account for.

[ QUOTE ]
I don't know what you mean by that... You don't know what your opponents upboard will be any more than you know what yours will be.

[/ QUOTE ]
Yes, you are right: you are just sampling from a blind deck. The probability is calculated as if the cards were dealt face down and then turned face up at the end.

7stud · #6 01-26-2005, 01:02 AM

Here is something I've been thinking about lately:

Raising to Limit the Field
---------------------------
Suppose this is the situation in a three handed $20-40 stud game, and p1 raises:

p1: Kc 9s Kh_______raise to $20
p2: Th Ts Ad___________________?
p3: 4d Jh Jd__bring in $5

Should p2 reraise in an attempt to drive p3 out of the hand?

Currently, p2's hand has a 31% chance of winning the pot:

[ QUOTE ]
7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
Kc Qd Kh 206651 41.33 293340 58.67 9 0.00 0.413
Ts Ad Th 154769 30.95 345222 69.04 9 0.00 0.310
Jd 4d Jh 138569 27.71 361427 72.29 4 0.00 0.277

[/ QUOTE ]
If p3 folds, then p2's chances of winning the hand increase approximately 10.5%:

[ QUOTE ]
7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
Kc Qd Kh 291679 58.34 208306 41.66 15 0.00 0.583
Ts Th Ad 208306 41.66 291679 58.34 15 0.00 0.417

[/ QUOTE ]
Is that enough? To proceed any further, I had to make some assumptions about the future betting(see the end for details). Based on the betting assumptions I made, the way I see it, these are p2's choices:

1) call and play the hand three way with a .31 chance of winning $449:

........Expected value = .31 * $449 = $139

2) or reraise and play the hand heads up with a .42 chance of winning $374:

........Expected value = .42 * $374 = $157

So, if p2's reraise succeeds in knocking out p3, then p2's $20 raise on 3rd street will increase p2's Ev by $18. That seems like a bad bet to me: p2 would be paying $20 for an $18 increase in profit.

-----------------------------------
-----------------------------------
Here are the betting assumptions I made, which I don't think are unusual:

If p2 doesn't reraise on 3rd street, I assumed this would be the betting:

antes: $9
p1: Kc 9s Kh _______raise to $20
p2: Th Ts Ad _________________call $20
p3: 4d Jh Jd __bring in $5______________call $15

3rd street total: $69
4th: $20 + $20 + $20
5th: $40 + $40 + $40
6th: $40 + $40 + $40
7th: $40 + $40 (one player drops out)
total=$449

If p2 raises on 3rd, and p3 folds, then I assumed this would be the betting:

antes: $9
p1: Kc 9s Kh_______raise to $20____________________call $20
p2: Th Ts Ad___________________ reraise to $40
p3: 4d Jh Jd__bring in $5______________________fold

3rd street total: $94
4th: $20 + $20
5th: $40 + $40
6th: $40 + $40
7th: $40 + $40
total=$374
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Any comments? Glaring errors?

bygmesterf · #7 01-26-2005, 01:24 AM

[ QUOTE ]

Any comments? Glaring errors?

[/ QUOTE ]

What kind of 20/40 game has only 9+5=$14 in antes??

Party == 8x2+5 = $21 -- Tight and boring
Normal == 8x3+5 = $29

I think raising to limit the field (which I think is an overated and often misapplied concept) balences out in a games with a reasonable ante

7stud · #8 01-26-2005, 02:21 AM

[ QUOTE ]
Suppose this is the situation in a three handed $20-40 stud game...antes: $9

[/ QUOTE ]
[ QUOTE ]
What kind of 20/40 game has only 9+5=$14 in antes??

[/ QUOTE ]

$3 ante/player X 3 players = $9. The $5 bringin is shown in the diagrams, and it is included in the 3rd street total along with the other bets.

I probably should have assumed 8 players with a $24 ante, which would add $15 to the total pot, so the calculations would be:

Based on the betting assumptions I made, these are p2's choices:

1) call and play the hand three way with a .31 chance of winning $449 + $15:

........Expected value = .31 * $464 = $144

2) or reraise and play the hand heads up with a .42 chance of winning $374 + $15:

........Expected value = .42 * $389 = $163

So, p2's $20 bet increases his profitability $19.

[ QUOTE ]
I think raising to limit the field (which I think is an overated and often misapplied concept) balences out in a games with a reasonable ante

[/ QUOTE ]

Since there is always some chance p1 and his King, does not have a pair of Kings, one could argue raising is either slightly wrong or very right. For instance, if p1 had a King high straight flush draw, which would also be a raising hand:

[ QUOTE ]
7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
Ks Qs 9s 159206 31.84 340782 68.16 12 0.00 0.318
Ts Ad Th 159182 31.84 340804 68.16 14 0.00 0.318
Jd 4d Jh 181596 36.32 318398 63.68 6 0.00 0.363

[/ QUOTE ]

and the p2's reraise succeeded in knocking out p3:

[ QUOTE ]
7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
Ks Qs 9s 209593 41.92 290382 58.08 25 0.01 0.419
Ts Ad Th 290382 58.08 209593 41.92 25 0.01 0.581

[/ QUOTE ]

p2's chances of winning the hand would vault from 32% to 58%, and the following increase in Ev:

1) call and play the hand three way with a .32 chance of winning $464:

........Expected value = .32 * $464 = $148.50

2) or reraise and play the hand heads up with a .58 chance of winning $389:

........Expected value = .58 * $389 = $225.50

In that case, p2's $20 bet increases his profitability $77--although I think the assumption about the total pot size would have to be lowered, so that is high.

grb137 · #9 01-26-2005, 04:02 AM

[ QUOTE ]

Any comments? Glaring errors?

[/ QUOTE ]

Couple comments = unless you're in a game of fish, no way do the jacks call that fall. I wouldn't play Js against Ks, and I certainly would play against Kings and someone who has called a raise from kings.

I am convinced that the raise is profitable assuming you know that the bring-in will fold. I love heads up. Heads up you can win many hands with two small pairs. Multi-way you really cant. Although your calculation shows the raise to be a losing play, if think its not quite right because it doesn't take into account those times when I fold before showdown (any open pair, apparent flushes, etc). It also doesn't take into account that I will fold the river if I didn't improve my pair.

I don't know how these factors can be incorporated into a calculation. I just know that there are tons of times where I fold before showdown when I'm chasin with a medium pair after having reraised to narrow the field. And I am a winning player, so it can't be that bad I guess.

BeerMoney · #10 01-26-2005, 03:25 PM

8 [img]/images/graemlins/spade.gif[/img] 7 [img]/images/graemlins/spade.gif[/img] 9 [img]/images/graemlins/diamond.gif[/img] vs Q [img]/images/graemlins/heart.gif[/img] A [img]/images/graemlins/diamond.gif[/img] K [img]/images/graemlins/club.gif[/img] with Dead Cards 3 [img]/images/graemlins/club.gif[/img] 7 [img]/images/graemlins/club.gif[/img] j [img]/images/graemlins/club.gif[/img] 3 [img]/images/graemlins/diamond.gif[/img]
7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
8s 7s 9d 253270 50.65 246677 49.34 53 0.01 0.507
Kc Ad Qh 246677 49.34 253270 50.65 53 0.01 0.493