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WHAT ARE THE ODDS?
I've seen a couple of questions on this forum asking about the odds of making specific hands in Texas Hold 'Em.
Since I've got time on my hands and I'm a maths nerd I'll have a go at answering any questions that people ask.
There are many web sites that give miscellaneous statistics, but as an added bonus, I'll try to "show my working" where applicable (like a good mathematics student!) which should give some clue how to work out other odds for yourself.
I give no guarantees about the accuracy of my answers but I am happy to debate the answers with anybody and, as I'll be showing how I calculated the figure, hopefully it should be clear how the answer was arrived at.
As a starter, I believe that a couple of nights ago Mr. Banin asked what the odds of flopping a royal flush are before any cards are dealt, i.e. before you know what your hole cards are.
Well, the odds are 649,739 to 1.
This is worked out as follows.
Your first hole card can be any one of the 52 cards in the deck. There are 20 cards that you could be dealt that would make up part of a royal flush, i.e. any card from ace to ten in each of the four suits.
The second hole card must be one of the 4 remaining cards of the same suit that is also in the range ace to ten and there are now 51 cards left in the deck.
The first flop card must be one of the 3 remaining cards that make up the royal flush from the 50 left in the deck.
The second flop card must be one of the 2 remaining cards that make up the royal flush from the 49 left in the deck.
The third flop card must be the final remaining card that makes up the royal flush from the 48 left in the deck.
So, the number can be calculated as 20/52 * 4/51 * 3/50 * 2/49 * 1/48 = 0.0000015307716932927.
(For those not familiar with this sort of notation this means 20 divided by 52 multiplied by 4 divided by 51 etc.)
If you divide this number into one, you get the odds, i.e. 1 in 649,740 or 649,739 to 1 if you prefer.
This information has no practical use that I can think of, but you may find it interesting. If you're among the majority that doesn't find it interesting, I didn't force you to read this!
Comments
What are the odds of Sky offering you Marks job?
Good post....
If you like numbers and are logical, you can work it out in less than 2 minutes.Same with the lottery when it first came out I knew the exact odds in a minute. 6 x5 x 4 x 3 x 2 x 1. divided by 49 x 48 x 47 x 46 x 45 x 44.
Im still not very good at poker though. Its a good job I can make a living betting on football. Happy number crunching.
Here is an example of a bad beat:
Player1 has AcTs
Player2 has 6h6s
The flop is 6dTh5s
All the money goes in.
What are the odds for the two hands Heads Up?
Well, the only way for Player1 to win is to make a better full house than Player2.
The turn must be a ten, an ace or a five.
If the turn is one of the 2 remaining tens, then the river can be the sole remaining (case) ten, one of 3 aces or 3 fives.
If the turn is one of the 3 remaining aces, then the river can be one of the remaining two tens or two aces.
If the turn is one of the 3 remaining fives, then the river can only be one of the 2 remaining tens.
So the odds are:
(2/45 * 7/44) + (3/45 * 4/44) + (3/45 * 2/44) = 0.016 approximately or 61 to 1.
The worst possible bad beat you can have in Texas Hold 'Em poker after the flop is where a person requires two perfect cards to make their hand.
To give an example,
Player1 has AhKh
Player2 has 2h2s
The flop is As2d2c
Then Player1 needs both aces to make better quads.
This would be 2/45 * 1/44 = 0.001 or 989 to 1.
I know that it is much simpler to use an poker odds calculator (available on many well know web sites) to calculate this but I thought that I'd calculate it manually as I regained my equilibrium. It's much more productive than smashing the laptop up.
Hi Tikay
The softwarestill not recognising your new hair style?
I did not win comp other night when i said it was 650,000, guess i was wide of the mark huh
LOL
Arguably, knowing odds is more important online because you are unable to make use of reads and physical tells in the same way as a live game.
I'm happy to try and help with any questions regarding the mathematics of poker.
have a look at the hand ive posted under bad beats.... i reckon the odds on turn and river astronomical..it was 9 players left on a final table
In Response to Re: WHAT ARE THE ODDS?:
hi,Your post was
"For all you guys that think bad beats only happen on Sky Poker !!!!!!
I was playing a tourny today on Party Poker and was 2nd chip leader. Dealt KK on the button. Shortstack under gun pushes and is called by another shortstack and then by chip leader making pot of about 4k. Guessing chip leader is chancing on it to put out both of them I stuck 8k in which he flat called. Flop was 8 4 5 rainbow. I thought if AA was about it would be with a shortstack so I pushed. And was immediately called.
Amazingly all three of them showed AQ not suited.
The turn came up with the case Q which was great for me !
Yes you guessed it .......the river was the case Ace "
There is only one possible way for you to loose after that flop, the case ace must come on the turn or the river. There is no straight, flush or trips options for any of them.
Given that we know 11 cards (8 hole cards plus the flop), the odds of the ace coming are 1/41 + 1/40. Pretty much 20 to 1. Of course you could always redraw with the king which would have been sweet but basically you are 92% to win outright.
You have another 2% chance to split it with runner, runner 67 too!
I'll go into more detail later to bore people!
In Response to Re: WHAT ARE THE ODDS?:
Hi Dazler,
Flattery will get you everywhere.
Personally, I think odds calculators are very useful.
I don't think that I'm allowed to give links to other web sites in this forum but if you type the words "poker odds calculator" into your preferred internet search engine you'll find some very good sites that allow you to select any hole cards (and flop & turn cards) you like and will give you the precise odds of each hand. This is a very useful tool for getting a feel for how strong your hand is.
Some examples of pre-flop odds that are frequently quoted on tv are:
Hand1 Hand2 Hand1 odds%: Hand2 odds%: Comment
===== ===== =========== =========== =======
KsKd Kh2d 95 5 Biggest possible pre-flop favourite
AsAc KsKc 83 17 Typical pair over pair
AhKh 2d2c 50 50 A "coin toss"
JhTh 2d2c 54 46 JT is better than AK against a small pair!
Ah2h 6s6c 34 66 Rag ace against a pair
AsKc Ad8h 74 26 A dominated hand
AsKc QhJd 64 36 Two overs v. two unders
As3c 8h7d 55 45 Pretty close!
KsKc KhQc 92 8 Easy for Tikay!
(Note that I have split the odds of the tie into the individual hands for simplification.)
There are even tools that give you the odds of your hand against a range of hands, e.g. say you have JJ and you put your opponent on AA, KK, QQ, AK, or AQ. The odds calculator tells you the average odds of your hand against all of these hands. This is a very powerful tool and much more useful than a straight odds calculator since it is generally impossible to put your opponent on an exact hand (unless it's me raising in which case you know I always have AA).
In general, for post flop calculating, just about everyone uses the "rule of four and two", i.e. each out that you have is worth 4% after the flop and 2% after the turn. Note that the 4% only applies if you are all-in on the flop and are guaranteed to see the turn AND river.
This is a reasonable rule of thumb and just about all we humans can cope with given a 10 second timer.
As I'll illustrate in another post, it's only a guideline though, and the number of "true" outs can influence the actual odds greatly.
Putting your opponent on a hand is by far the most important aspect of this. I particularly like a statement by the great Jesse May (possibly quoted from someone else) about the dangers of the "creative put". i.e. the habit of putting someone that has gone all in on a hand that you can beat. Basically, he said, "I go through all the hands that he might possibly have that I can beat. Then, if I can't think of any of those, I put him on a complete bluff and call anyway"!
I hope that this is useful. Please feel free to ask anything else.
For those interested in the hand featured on last night's show where I e-mailed in and said that CHRISG79 made a good fold, here are some more thoughts on the hand. In my rush to send the numbers into Ed in a format that could be used on the show I may not have given as good an analysis as I should have at the time.
(Incidentally, trying to provide some numbers in a reasonably quick time made me appreciate how quickly Ed works on the show - having to provide the numbers in "real time" without the aid of a computer).
In fact, I made a mistake relating to pot odds that I am quick to criticise other people for, so I must apologise.
Pot odds are obtained by dividing the amount of chips already in the pot by the amount that you have to put in to call, you DO NOT count your chips as being already in the pot.
As an example, if there are 2000 chips in the pot and your heads up opponent puts 2000 in (i.e. bets the pot) you have to call 2000 into a pot of 4000. In this case you are getting pot odds of 2 to 1 NOT 3 to 1.
The featured hand show the importance of "putting someone on a hand" when calculating your hand odds.
The arguments below are based purely on the maths of the situation rather than any meta-game considerations relating to laddering or the size of FUIFUI's remaining stack (apart from the fact that he has CHRISG79 out-chipped) or any other issues.
Details of the hand are:
CHRISG79 - Hole cards: KdQs Stack: 8080
FUIFUI - Hole cards: AsTs Stack: 18942 +300 already in the big blind.
Blinds: 150/300
Six people at the table.
CHRISG79, on the cut-off, raised to 900.
FUIFUI called in the small blind and the big blind folded.
This meant that there was 2100 in the pot pre-flop.
f=9s9dTh
The odds after the flop for the actual hands are 66% to 34% in favour of FUIFUI (about 2 to 1).
If FUIFUI's hand was As9h the odds are: 85% to 15% (almost 6 to 1)
If FUIFUI's hand was 5h5c the odds are: 52% to 48% (pretty close to even money).
This shows an enormous difference between the odds depending on FUIFUI's hand.
It also shows how the "rule of four and two", i.e. giving yourself 4% for every out that you have after the flop (and 2% after the turn) is only a rule of thumb. However, in online poker, with a fast running timer, this rule is as good as anyone can reasonably do (unless you're Andy Bloch or Chris Ferguson, I guess).
After the flop, FUIFUI checked, CHRISG79 bet 900 and FUIFUI check raised to 2400 and CHRISG79 folded.
The chip counts after FUIFUI's check raise are:
Chips in the pot: 5400
CHRISG79: 6280
FUIFUI: 15642
We can discount fold equity here, i.e. we should assume that FUIFUI is always going to call if CHRISG79 shoves all his chips in. This means that we can treat the pot odds exactly the same as if FUIFUI had gone all-in after CHRISG79 bet.
If this were the case, there would be 9,280 chips in the pot and CHRISG79 would have to call 6280 more.
This gives pot odds of about 1.5 to 1 (not 2.5 to 1 as I said in my e-mail, ahem).
The following comments are more related to poker strategy than pure poker maths but show how the two are linked.
The key to the decision, I think, is that FUIFUI check-raised on the flop in a position where he knew that CHRISG79 might have felt nearly pot committed. This, to me, indicates a very strong hand (or a willingness to gamble). Therefore it might be reasonable to put him on a 9 and means that only the 4 jacks are true outs.
The only hand that FUIFUI could have that would make it reasonable for CHRISG79 to go all-in is an underpair to the board (or complete air).
Therefore, for me, this was a GOOD FOLD.
Personally, I think that the Sky show could be improved by enhancing the hand display to include the odds of each hand on the screen as done in many televised live games - but that's just me, I guess. It would save the presenters having to try to work things out on the fly and because of the large number of hands featured would be highly educational. Anyone not interested in the odds could easily ignore them.
Feel free to ask questions ay any time.
If I don't know the answer, I'll bluff.
i have worked it out and its defo 649,740 to 1..........
lol, n1.
1 in 649,740 or 649,739 to 1!!!
Those two figures are two different ways of expressing exactly the same odds!
Yes, I flopped one once on here - but I've played a lot of hands!
A hand from a DYM tournament this morning.
Blinds are 75/150 and I still have 1380 chips.
Two people limp and I make up the sb with KcQc.
The big blind just calls.
f=Tc9s3c and I check.
The big blind minimum bets and everyone folds back to me and I shove the lot in.
The big blind calls with Ts7h because he has top pair.
So at this point he is ahead because he has top pair.
I have two overcards, a flush draw and a gut shot straight draw. Apart fom flopping quads, I could hardly wish for more.
I have 9 clubs, 3 jacks, 3 queens and 3 kings to hit. That is 18 outs with 2 cards to come. Using the rule of "4 and 2" I would be about 72% favourite - in reality I'm about 61% favourite because he has 5 outs to re-draw to 6 of my outs (i.e if I hit a K or Q and he hits a T or 7 - he still wins).
Sadly, I hit none of my cards and I am out and can only say "gg".
So maths says I am right to go all in (with the added equity that he might fold too) but 4 times out of ten he is still going to win (despite being ahead already!).
That is how you can be behind and ahead at the same time!
I have Qd2d
Opponent has Js3s.
We both limp pre-flop.
f=Qs2sJh
We manage to get it all in on the flop.
I have two pairs and he has middle pair with a flush draw.
t=Th
r=Jd
I lose again - sniff.
What are the odds?
Well, after the flop, I am also exactly 60% favourite but he has such a big hand that he is never going to fold.
He doesn't expect that his three 3's aren't outs and heads up you can't really expect your opponent to have such a big hand.
If I just had a queen, we would be 50% each - the proverbial coin-toss!
"hi guys this is one of my first posts on here, i actually cant believe whats happend. i was cruising in the open managed to get a good stack together, as the tournys moving im finding myself losing chips from blinds as im playing good solid poker. 69 players left, i have around 24 BB's blinds are at 200/400. i pick up QQ in early position i raise to 1600 the chip leader at the table has around 6k more than me he calls from the BB, flop comes down 458 he bets 2,500 at me i shove for my remaining 9k of chips he snap calls me with 87 the turn comes down an 8 then the river comes down another 8 lol i cant believe it such a sick beat and he hits the 2 remaining 8's what are the actual odd's of him doing this hitting running 8's and the remaining case 8?"
The odds of making quad 8's are 2/45 * 1/44. This equates to 989 to 1.
Of course, there are other ways for the 87 to win.
Any 8, 7 or 6 gives him the win which is 9 nine outs.
Using the "rule of 4", we would say that this is approximately a 36% chance to win.
Individually, there are re-draws to most of these outs:
If one of two 8's hits, one of the two remaining Q's will still win it for ginge999.
If one of the three 7's hits, one of the two remaining Q's or any of the six remaining 4's or 5's will win it for ginge999.
If one of the four 6's hits, one of the remaining 3 7's will split it for ginge999 (and vice versa).
Therefore, the true odds are:
For the 87 to win:-
(2/45 * (1 - 2/44)) + (3/45 * (1 - 8/44)) + (4/45 * (1 - 3/44)) * 2 = 35.96%
For the split pot:-
4/45 * 3/44 * 2 = 1.21%
(the final two in the calculations is because the two cards can come in either order)
Therefore we can see that the "rule of 4" is remarkably accurate in this case.
In general the "rule of 4" works pretty well because the true figure for each "out" is 4.5% but discounting it to 4% gives a margin for the re-draws that the opponent has. As you can see from other calculations above, it isn't always so accurate but it is a very reasonable approximation in most cases.
p1: QhTc
p2: 9c9d
f=KdJs9h
Well, these are two big hands. p1 has flopped the nuts and p2 has flopped the bottom set.
p1 is winning but what are the odds of p2 winning or it being a tie?
p2 will win if he makes a full house or quads.
There are 7 cards for p2 to improve on the turn (in which case the river card is immaterial) and another 10 cards to improve on the river if he doesn't on the turn.
Therefore the odds are 7/45 + (38/45*10/44) = 34.75%
The pot will be split if they both make the same straight.
This will happen if a Q or T comes on the turn and the other card comes on the river, this makes 6 cards on the flop and 3 cards on the river.
Therefore the odds are 6/45 * 3/44 = 0.91%
What are the maximum number of outs that you can have on the river? (if you're not actually winning already!).
Well, the maximum outs would be if you had an opened ended straight flush draw with 4 overcards to the pair on the board (two of which are a pair) against an underpair.
An example is:
p1: 2c2s
p2: KhQh
board=ThJhTs8s
In this case, p2 wins in the following cases:
any of eight hearts to make a flush (i.e. not the 2h)
any of three (non-heart) K's to make a better two pair
any of three (non-heart) Q's to make a better two pair
any of three (non-heart) A's to make a straight
any of three (non-heart) 9's to make a straight
any of three J's to make a counterfeit two pairs
either of two (non-heart) 8's to make a counterfeit two pairs
This is 25 outs and the odds are 25/44 = 57% (not too shabby, really!)
The first is from another 7pm deepstack.
p1: KhKs
p2: 6c6d
f=5d4d3d
There had been some pre-flop raising and re-raising.
Anyway, we can see that p2 has flopped an open-ended straight flush draw. If you add in the further two outs for making a set of 6's, this means that he has 17 outs after the flop (nine diamonds, three more 7's, three more 2's and two 6's).
The rule of 4 would say that he is 68% to win the hand. In fact, due to the number of re-draws for some of those outs, he is actually about 60% favourite with another 1.2% chance of a split. (Detailed maths available on request!).
The two players got it ai on the flop and the K's held up.
Later that same evening in a Bounty Hunter, I held 6c9s in an unraised big blind.
p1: 6c9s
p2: Kc2c
f=5c7c8c
In this case, I hold the nut straight and an open-ended straight flush draw.
Sadly my opponent already has a flush which makes my straight redundant and most of my flush draws redundant too.
In fact, only the straight flush will win the hand for me, so I have only 2 outs.
This means that I have only a 8.65% chance of winning the hand.
That's quite a difference to the hand above even though I have flopped a seemingly stronger hand.
These two hands illustrate very well, I think, the importance of putting your opponent on a hand when calculating your odds!
P1: AdAc
P2: Jh2h
f=6hJc8h
So, what are the odds?
P1 has the best hand so far but P2 has 14 outs. If he hits one of the nine hearts, P1 has no redraws (unless it is the ace of hearts in which case he has 10 re-draws). If P2 hits one of the two jacks, P1 has a re-draw to one of the two remaining aces. If P2 hits his 2, then P1 has eight re-draws to make a better two pair or a set.
The rule of 4 says P2 is 56% favourite. The actual maths say that this is almost exactly 50/50 - another proverbial coin-flip.
I'm going to cover something slightly different to the normal odds questions here.
Last night on the Bounty Hunter coverage, there was a hand where four of the players had pocket pairs - 55, JJ, KK, AA.
Trevor and Richard decided that the odds of this were 16 * 16 * 16 * 16/1 - i.e 65,536 to 1.
This would be the correct answer if there were only four players at the table.
However, at a 6-seater table, you have to divide these odds by the number of ways of choosing four players from six.
There are 15 combinations (to use some mathematical jargon) of 4 from 6.
Therefore the odds are actually 4,369 to 1 - still pretty unlikely!
You need to hit both of your cards in the first three cards dealt.
Therefore, the odds are 2/50 * 1/49 * 3 = 0.00245 or 407/1.
The final three in the calculation is because there are 3 ways of choosing 2 cards from the 3 dealt cards.
So, this isn't quite as rare as you might think.