7 card stud - Calcluating Probability for improving 1p - please help

[Pok Jung Kee]

Hi poker gods,

I am confounded by this one. On Mike Caro's site, assuming 0 dead cards (throughout the hand), he states that the probability for making 1p into 2p or trips is 52%, 60% if you consider improving to a boat. (I don't even care about that extra 8% right now)

When I calculate it, I come up with a number better than 100%, and I know from experience that the number is closer to 50%. LOL. Please help me. What is my fundamental flaw in calculating probability here.

Let's assume the following: 8 person table, headsup play from 4th street forward. Starting hand is AA9. (This could be any pair, any kicker)

3rd) So with AA9, I have 7 cards to improve and 10 unseen cards, so 7/42 to improve.
Any unhelpful card falls, say a 6.
AA96
4th) So with AA96, I have 3 more cards to help, so 10 to improve, and 12 unseen cards, so 10/40 to improve.
Any unhelpful card falls, say a 4.
AA964
5th) So with AA964, I have 3 more cards to help, so 13 to improve, and 14 unseen cards, so 13/38 to improve.
Any unhelpful card falls, say a 7.
AA9647
6th) So with AA964, I have 3 more cards to help, so 16 to improve, and 16 unseen cards, so 16/36 to improve.
And wouldn't you know it, on the river I do NOT catch any of those 16 cards. GRRRRR.

I add these fractions, since it's an "OR" condition I improve if A or B or C or D occurs, and when I add those numbers, (I round the denominators to 39 to simplify)
(7+10+13+16=46) so about a 46/39 chance = I should improve 120% of the time!!!

People steamroll my pocket Aces all the time in 7CS.

Odds-masters, please help, I know this is something small and stupid, but perhaps others make the same fundamental mistake in their calculations.

thank you,
Pok Jung Kee

[AaronBrown]

You did nothing stupid, just made two minor errors. There's also an important nuance.

The biggest problem is adding the odds. Using your numbers, you have 1 chance in 6 of hitting on the fourth card, 1 in 4 of hitting on the fifth. The chance of hitting on one or the other is 1/6 + (5/6)*(1/4). You have to multiply the 1/4 by the chance of not hitting on the first card or you'll double count. If you do the entire calculation this way, you'll get 77% instead of 120%. In general, it's easier to compute the odds of not making the hand on each card, then you can multiply them together.

The nuance is you are assuming none of the cards you see in opponent's hands help you, even retroactively. For example, someone might fold with 7 showing, and you get another 7 later. That would change the calculation.

I think it's better to compute the odds ignoring your opponents' hands. That gives you the unconditional probability of making your hand. On average, your opponents will get the average number of cards that help you. In a game, of course, you have to compute the conditional probability.

The smaller mistake you made was starting with 7 cards that help you. With AA9 there are 2 Aces and 3 nines for only 5 helpful cards. So the way I do it you have: 5/49, 8/48, 11/47 and 14/46. Figuring the odds of not doing it is (44/49)*(40/48)*(36/47)*(32/46) = 40%. So the odds of hitting are 60%. This includes full houses and quads (but not straights or flushes). There's a 40% chance of ending up with no paired cards other than your pocket Aces; and a 60% chance of improving by some kind of pairing.

[Pok Jung Kee]

Thank you for the instant help.
I see the value in calculating the chances of NOT catching.
I double checked my work calculating it in both directions, and I get 69.4% with zero dead cards both times!! Nice, it worked.
Thank you!!!

I would like to test my "probability muscle" again with your help.

Now, let's try add "reality" into this IDEAL-WORLD scenario of 69.4%.

This formula assumes deadcards=0 for all 4 streets. What I want to know is:

3) What is the prob. that your AA9 on 3rd has no dead cards?
4) What is the prob. that your AA96 on 4th has no dead cards?
5) What is the prob. that your AA963 on 5th has no dead cards?
6) What is the prob. that your AA963K on 6th has no dead cards?

How would you set that formula up?

This way you can add this "average probability" into each street for a real-world number % of what to expect for going forward with your Aces, and hoping (praying) to catch 2p or better against someone who apparently caught 2p from the start like 4s/2s, and is challenging you that you are still on your Aces.

I would like to know how to calculate this precisely. So in my rough estimate, I removed 1 dead card from each street, and come up with 60.3%. This is almost Mike Caro's number of 60.05%. http://207.176.137.71/mcu/tables/Table28.asp (He does not state the variables under which this 60.05% figure is calculated.)

I came up with .55/.72/.92/.98 dead cards to remove from each street, by calculating the odds of nobody catching a dead card against you. I did it this way, it make be incorrect:
3rd street (5 cards hurt you): (44/49 * 43/48 * 42/47 * 41/46 * 40/45 * 39/44 * 38/43) = 45% live, so 55%. So I remove .55 dead cards
4th street (8 cards can hurt you):
(41/49 * 40/48 * 39/47 * 38/46 * 37/45 * 36/44 * 35/43 * 34/42) = 21% live, so remove .79 dead cards
5th street (11 cards can hurt you):
(38/49 * 37/48 * 36/47 * 35/46 * 34/45 * 33/44 * 32/43 * 31/42 * 30/41) = 8% live, so remove .92 dead cards
6th street (14 cards can hurt you):
(35/49 * 34/48 * 33/47 * 32/46 * 31/45 * 30/44 * 29/43 * 28/42 * 27/41 * 26/40) = 2% live, so remove .98 dead cards

This gives me a real-world % of 61.1%, which is MUCH lower than 69.4%, AND I did NOT round off the numbers, (knowing cards must be whole numbers), but I used fractions to obtain an average percentage.

(I realize when I am all done, I have to go back and calculate what are the chances of my opponent improving their low 2p against my Chasing high 1p), and subtract that from my calculations. But I am not too worried about that counter-case right now. heh.)


thx,
Pok Jung Kee

[AaronBrown]

I'm not trying to discourage your study of probability, but this is not a very useful question.

When you have AA9, you want to know your probability of improving. So you look around and see how many Aces and 9's are dead before doing the computation. It doesn't make much difference what the probability of someone else getting an Ace or 9 next round. You don't have to figure that into your calculations until next round, and by then you'll know the answer.

However, it's easy to calculate. The probability of someone else getting one of your useful cards is exactly the same as you getting it.