Math problem with playing cards

[dogooder]

Originally posted by: dullard
3 of a kind with three cards
1) First card doesn't matter. Whatever number the card is, will determine what the other two must be.
2) 51 cards remain. Three of them match your first card. Thus your odds are 3/51 to get the first two to match.
3) 50 cards remain. Two of them match your first two cards. Thus your odds are 2/50 to get the third card to match.

Overall odds: 3/51 * 2/50 = 1 out of 425.

4 of a kind with five cards
1) There are 672 possible four of a kind hands
2) There are a total of 2,598,960 possible hands.
3) 672/2,598,960 = 1 out of 4165 odds of getting a four of a kind.

Using the same reasoning, for 4 of a kind in 5 cards:

If we get the 4 cards of the same value first, and the other card last, then the probability is 52/52 * 3/51 * 2/50 * 1/49 * 48/48 = 1/20825. But the other card could be first, second, third, fourth, or last. So the probability is 5/20825 = 1/4165.

Getting a three of a kind is almost ten times more likely to happen.

Edit: I'm slow.. clicked quote before I saw his edit.

 

[ActuaryTm]

Originally posted by: Tommunist
if the question is what are the odds that at least ONE person out of the 5 will end with the desired result will have different probabilities.

Precisely.

 

[amoeba]

dullard's analysis is correct

 

[ActuaryTm]

Originally posted by: dullard
You quoted my entire posts (and you didn't edit it). Thus, you can see that it was not added during the edit. Look at your 9:16 post.

You've misunderstood the meaning of "added duing the edit". FuseTalk cascades edit changes down to quotations by other users; thus, as I was posting, the edit was made, and the insertion appeared without me first seeing the result (e.g., I hit reply before the edit, then submitted afterwards).

It's fairly common, and not a bug of course (it's a feature).

 

[dullard]

Originally posted by: ActuaryTm

Originally posted by: Tommunist
if the question is what are the odds that at least ONE person out of the 5 will end with the desired result will have different probabilities.

Precisely.

In this case, if you use more players, the "odds of at least one person" getting the specified hand is higher than the "odds of you personally" getting the specified hand. That is true of the 3/3 and the 4/5. Thus, both situations get easier if you use the "odds of at least one person" definition. So as a comparison of one situation to the other, why complicate things? It doesn't change the comparison that one is 10 times harder than the other.

 

[Tommunist]

Originally posted by: ActuaryTm

Originally posted by: Tommunist
if the question is what are the odds that at least ONE person out of the 5 will end with the desired result will have different probabilities.

Precisely.

it's also a more complicated problem that i don't feel like thinking about. but my gut tells me the 3 of a kind is still the most likely

 

[dullard]

Forget it.

 

[ActuaryTm]

Originally posted by: dullard
So as a comparison of one situation to the other, why complicate things?

I suppose that is a direct result of my profession. I always find it easier to assume the worst case first.

 

[ActuaryTm]

Originally posted by: dullard
So are you saying I lied about not editing that portion?

Good grief. Is this really this hard to comprehend?

No, no lying. *I* didn't see the addition until *after* I submitted, even though it was added graciously for me in my quotation of your words by FuseTalk. Better? (or at least, clearer?)

If anything, the fault is mine (for taking so long to respond after hitting reply).

 

[amoeba]

now that the OP's question has been answered, I'll threadjack with another poker quiz.

you are playing no limit texas holdem. everybody at the table has very deep stacks to where the blinds are almost insignificant.

UTG player moves all in preflop. You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

 

[dullard]

Forget it.

 

[SuperPickle]

Thanks for the responses, fellas. :beer:

 

[ActuaryTm]

Originally posted by: dullard
No, the fault is yours for not seeing it originally, as it was not edited in.

(just dying laughing)

We'll define that as event A, and assign it a probability 10 times less likely than the "four of a kind" event.

 

[ActuaryTm]

Originally posted by: amoeba
You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

Know, as it certain, or "know" as in feel?

 

[scorpmatt]

Originally posted by: amoeba
now that the OP's question has been answered, I'll threadjack with another poker quiz.

you are playing no limit texas holdem. everybody at the table has very deep stacks to where the blinds are almost insignificant.

UTG player moves all in preflop. You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

call

 

[Tommunist]

Originally posted by: ActuaryTm

Originally posted by: dullard
So as a comparison of one situation to the other, why complicate things?

I suppose that is a direct result of my profession. I always find it easier to assume the worst case first.

worst case != most complicated case.

you can make your life a lot easier by starting with simplier versions of whats going on and building up if required. as someone who presumably studied mathematics i find it suprising you don't approach problems in this manner.

 

[ActuaryTm]

Originally posted by: Tommunist
worst case != most complicated case.

I should have taken greater care in defining worst case to be "worst financial" case.

My apologies.

 

[amoeba]

Originally posted by: ActuaryTm

Originally posted by: amoeba
You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

Know, as it certain, or "know" as in feel?

you are certain.

 

[amoeba]

Originally posted by: scorpmatt

Originally posted by: amoeba
now that the OP's question has been answered, I'll threadjack with another poker quiz.

you are playing no limit texas holdem. everybody at the table has very deep stacks to where the blinds are almost insignificant.

UTG player moves all in preflop. You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

call

gonna need some math here. its not as simple as it seems.

 

[Viper GTS]

Originally posted by: amoeba
everybody folds to you and you look down and see that you hold KK.

should you call or fold?

PokerStove says you have 47.3% pot equity vs 52.7 for him.

If you call that consistently you'll lose all your money.

Text results appended to pokerstove.txt

154,107,360 games 0.265 secs 581,537,207 games/sec

Board:
Dead:

equity (%) win (%) / tie (%)

Hand 1: 52.7071 % [ 00.49 00.04 ] { AA-KK, AKs, AKo }
Hand 2: 47.2929 % [ 00.44 00.04 ] { KK }


---

Viper GTS

 

[RaynorWolfcastle]

Originally posted by: amoeba
now that the OP's question has been answered, I'll threadjack with another poker quiz.

you are playing no limit texas holdem. everybody at the table has very deep stacks to where the blinds are almost insignificant.

UTG player moves all in preflop. You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

Assuming, you can't get any read off him, KK is roughly a 2:1 favorite to AKs. KK is about a 4:1 dog to AA and it's essentially a split if he has KK as well.

So you have roughly:
33%*66% + 33%*20% = 28.3% chance of winning
33% chance of tying
33%*66% + 33%*80% = 48% chance of losing

With that said, I personally, have never successfully laid down KK against AA (it's only happened a couple of times to me). Even pros will tell you that this is probably the hardest lay down you can make.

BTW, who the hell would push all-in preflop UTG with a deep stack looking with any of these hands?!? Much more likely you'd see a couple of raises and reraises before somebody pushed all-in and then it would be much harder to tell what the other person is playing with

 

[amoeba]

Originally posted by: Viper GTS

Originally posted by: amoeba
everybody folds to you and you look down and see that you hold KK.

should you call or fold?

PokerStove says you have 47.3% pot equity vs 52.7 for him.

If you call that consistently you'll lose all your money.

Text results appended to pokerstove.txt

154,107,360 games 0.265 secs 581,537,207 games/sec

Board:
Dead:

equity (%) win (%) / tie (%)

Hand 1: 52.7071 % [ 00.49 00.04 ] { AA-KK, AKs, AKo }
Hand 2: 47.2929 % [ 00.44 00.04 ] { KK }


---

Viper GTS

using poker stove is kind of cheating but yes the correct answer is fold.

 

[amoeba]

Originally posted by: RaynorWolfcastle

Originally posted by: amoeba
now that the OP's question has been answered, I'll threadjack with another poker quiz.

you are playing no limit texas holdem. everybody at the table has very deep stacks to where the blinds are almost insignificant.

UTG player moves all in preflop. You know 100% that he does this with either AA, KK, or AK.

everybody folds to you and you look down and see that you hold KK.

should you call or fold?

Assuming, you can't get any read off him, KK is roughly a 2:1 favorite to AKs. KK is about a 4:1 dog to AA and it's essentially a split if he has KK as well.

So you have roughly:
33%*66% + 33%*20% = 28.3% chance of winning
33% chance of tying
33%*66% + 33%*80% = 48% chance of losing

With that said, I personally, have never successfully laid down KK against AA (it's only happened a couple of times to me). Even pros will tell you that this is probably the hardest lay down you can make.

BTW, who the hell would push all-in preflop UTG with a deep stack looking with any of these hands?!? Much more likely you'd see a couple of raises and reraises before somebody pushed all-in and then it would be much harder to tell what the other person is playing with

that is true but you must think on the 2nd level that AK appears 16 ways while AA appears 6 ways. so in that sense its much closer.

But then you must consider the third fact that you hold 2 Ks in your hand thus the ways of AK happening diminishes to 8 ways and thus it becomes a fold.

If I had said he only does this with AA, KK or AQ. I believe call would be correct.

 

[Viper GTS]

Originally posted by: amoeba
If I had said he only does this with AA, KK or AQ. I believe call would be correct.

Yup, by a bigger margin too (roughly 56/44).

BTW how can you do this accurately without using simulation?

You'd have to consider every possible deck configuration, simulation is the only way that's even remotely feasible.

Viper GTS

 

[RaynorWolfcastle]

Originally posted by: amoeba
that is true but you must think on the 2nd level that AK appears 16 ways while AA appears 6 ways. so in that sense its much closer.

But then you must consider the third fact that you hold 2 Ks in your hand thus the ways of AK happening diminishes to 8 ways and thus it becomes a fold.

If I had said he only does this with AA, KK or AQ. I believe call would be correct.

I used pocketfives to calculate the probability of each event against KK, so all this is taken into account, which is why I get roughly the same results as Viper GTS .
BTW, he probably has better odds with AQs vs KK than AKs vs KK since the K is essentially dead to him with AKs vs KK