poker percentages/pot odds?

Question

what are the odds of three out of the five community cards being hearts?

Answer 1

If you meant AT LEAST 3:



Disregarding your hole cards:

(13 * 12 * 11) / (52 * 51 * 50) +

3 * (39 * 13 * 12 * 11) / (52 * 51 * 50 * 49) +

6 * (39 * 38 * 13 * 12 * 11) / (52 * 51 * 50 * 49 * 48)



= 9.2767%



But if you meant EXACTLY 3:

(5 c 3) * (39p2 * 13p3) / (52 p 5) = 8.1543%



(note: 52p5 is a shorthand for 52*51*50*49*48)



Alternative scenario #1 -- both of your hole cards are hearts:

AT LEAST 3:

(11p3) / (50p3) +

3 * (39 * 11p3) / (50p4) +

6 * (39p2 * 11p3) / (50p5)

= 6.4%



EXACTLY 3:

10 * (39p2 * 11p3) / (50p5) = 5.77%





Scenario #2 -- 1 of your hole cards is a heart:

AT LEAST 3:

(12p3) / (50p3) +

3 * (38 * 12p3) / (50p4) +

6 * (38p2 * 12p3) / (50p5)

= 8.2247%



EXACTLY 3:

10 * (38p2 * 12p3) / (50p5) = 7.3%





Scenario #3 --- neither of your hole cards is a heart:

AT LEAST 3:

(13p3) / (50p3) +

3 * (37 * 13p3) / (50p4) +

6 * (37p2 * 13p3) / (50p5)

= 10.3%



EXACTLY 3:

10 * (37p2 * 13p3) / (50p5) = 9%





If you need me to explain the method, just say so. There is also another method.



**EDIT**

Your wish is my command, pdq! But do not mistake this as an admission of being your female dog!



The reason you can't just do:

10 * (11p3 / 50p3)

is because not all ways of getting 3+ hearts are equivalent, there would be an error resulting from using the same denominator terms too many times and using too few terms in the denominator (getting a non-heart will change the denominator for the next heart, and will also insert an extra term into both the numerator and denominator). Doing the above pretends HHH is the same as XXHHH, which it is not. (11*10*9)/(50*49*48) is not equal to (39*38*11*10*9)/(50*49*48*47*46)



There are 3 different sets of permutations:

A) you can accomplish 3 hearts on the flop

B) you can accomplish them by the turn

C) you can accomplish them by the river



You'll notice my 3 lines of work correspond to those 3 sets. There is only 1 way to get them all on the flop: HHH. There are 3 ways to get them by the turn: HXHH, HHXH, and XHHH. There are 6 ways to get them by the river. Mathematically, every individual permutation belonging to the same category is equal to one another. For instance, P(hxhh) = P(xhhh). So that's why you can just multiply by 3 and 6.



As a way to check my work, notice: P(exactly 3) + P(exactly 4) + P(exactly 5) = 6.4%

(using my method for exactly N hearts)

And my method for exactly N hearts is simple: there MUST be exactly 5-N non-hearts. Those appear in my answer as the "39p2" aka 39*38. And all 5 cards must be accounted for, so there has to be 5 terms in the denominator (so you go from 50 to 46, hence 50p5).



And actually, look again: James' answers agree with mine, not yours :P

Answer 2

It depends on a few things. Do YOU already have 2 hearts in your hand? Do you have ZERO hearts in your hand? That makes a big difference.



I don't feel like figuring out both, so I'm going to assume that the only reason you'd ask this question is if you already had 2 hearts in your hand.



OK - first, as someone suggested, let's figure out the odds of making the flush in the first 3 cards. There are just 11/50 hearts left in the deck if you have 2 hearts. (PLEASE don't listen to any moron that tells you that it matters how many players there are at the table! Unless you specifically know that one or more of these people had one or more hearts, proceed as if all 50 cards besides your 2 cards are UNKNOWN. Why? BECAUSE THEY ARE UNKNOWN!!!)



So for the first of 3 cards on the flop, it's a 11/50 chance. For the 2nd card, assuming the 1st is a heart, now there is a 10/49 chance. The 3rd card is a 9/48 chance. This is approximately a 1 out of 119 chance to hit it on the first 3 cards. (Around a 0.84% chance.)



Now - you're wondering about the chances by the time all 5 cards have hit. Think of all 5 cards as being cards A, B, C, D, and E. The 3 hearts can be either ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, or CDE. That's 10 different ways that at least 3 out of the 5 cards will be hearts.



So in FIVE cards, you have 10 times the chance to make a flush. That means you have an 8.4% chance, or slightly better than a 1 out of 12 chance.



By the way - this really doesn't have anything to do with pot odds. If someone raises pre-flop and you're sitting with an 8-3 of hearts, you shouldn't be calculating the pot odds that you can make your flush. You should simply be mucking your hand. Pot odds come more into play when you already see the flop. If you were Big Blind and you were allowed a free ride into the hand with 8-3 of hearts, and then the flop hit 2 hearts, THEN you could start carefully thinking about your pot odds of getting your flush. (I say "carefully" because you could always end up against a bigger flush. It would still be a "proceed with caution" sort of hand.)



***EDIT***



huckleberry beat me to the post. The scenario I described above is most like his Scenario #1 where you start with 2 hearts. He came up with 6.4% as compared to my 8.4% figure. I'm not sure where I went wrong, but I would trust huckleberry's math over my math any day of the week!



huckleberry - if you have an extra moment to spare, can you easily see where I went wrong in my calculations? I know I didn't differentiate between getting "at least" 3 hearts and "exactly" 3 hearts, but I still can't see how I can be so far off in my answer.



If you get a minute, great. If not, no worries.



(***I see James got similar numbers to mine.)



***RE-EDIT***



I bow down to you, huckleberry! Thanks so much for taking the time to explain.



It's time for me to get back to math class! Thanks again.

Answer 3

Well the chances of getting 3 hearts and 2 non hearts in a specific order (for example (HHHNN) where H is heart and N is non heart, is given by:

(13/52)* (12/51)* (11/50)* (39/49)* (38/48)

= 2543112/311875200

And there are 10 possible combinations of 3 hearts and 2 non hearts (5C2)

so the chances of exactly 3 hearts coming out is:

= 2543112*10/311875200





= about 1/12 or 0.082



This is the chance of exactly 3 hearts and 2 non hearts coming out, if you do not know what any of the player's cards are.

Therefore, the chances of the community cards having exactly 3 of ANY suit is 4/12=1/3



These odds will be modified by the extra information you have when you know the suits of the cards in you own hand:



If you are holding 2 hearts in your hand, the chance of exactly 3 hearts coming out on the board will decrease to about 1/17



If you hold 2 hearts in your hand, the chances of flopping a flush (meaning, to make it on the first 3 community cards alone) are 1/119

Answer 4

The odds of the first card being a heart is 13/52, the second 12/52, the third 11/52. Multiply those together and you get the odds of three hearts out of three cards.



Now someone can figure how to handle the last two cards.

Answer 5

There are a few problems with your logic. First of all, there are only 12 outs (the 10's are not outs because 10's doesn't beat Q's). Next, the x4 rule is the probability that one of your outs will appear in the next 2 cards. So you are correct that you have roughly a 48% chance of beating QQ. However, that does not take into account the probability that the QQ gets improved. For example, if another Q hits then your K's are no good. If 2 Q's hit, your flush is not good. Or if a Q and 2, 4 or 10 hit your flush is also no good. The odds calculator takes all that into account. Another good rule is that a pocket pair against 2 overcards is usually close to 50-50 (for example QQ vs. AK). In this case there is only one over card (K), but I guess the flush draw helps even things out. To summarize, the x4 and x2 rules just give you the odds of you improving your hand. It does not take into account the odds of your opponent improving his hand. Hope this helps.

Answer 6

The odds of making a flush by the river after being dealt two suited cards is 6.25%

Check out some more interesting poker odds here:

http://www.areyouapokerfish.com/interestingodds.html

Answer 7

Your odds of making a flush by the river are 15:1.

Your odds of flopping a flush are 118:1.



Learn more pot odds here:

http://www.areyouapokerfish.com/outsodds.html



Learn about Implied Pot Odds here:

http://www.areyouapokerfish.com/impliedodds.html

Answer 8

Here is a poker odds calculator for you to help out...I use it all the time...

http://www.abonuscode.com/?cat=93

Answer 9

I use this poker odds calculator.. http://www.pokercalculatoronline.com/

Answer 10

no one can possibly know as it depends on how many people are at the table etc. without taking into account discarded cards it remains unknown. and your question has nothing to do with pot odds