Answers to all odds questions!

[Demiparadigm]

How to calculate odds:

Odds are determined based on probability. For example when you flop four to the flush the probability of you making the flush on the next card is 9/47 (9 remaining cards of your suit out of 47 remaining unseen cards.) You can convert this to a percentage by dividing 9 by 47 and get 19.15%.
Odds are usually stated as the odds against something happening. So, since there are 38 cards that won’t help you and 9 that will, your odds are 38:9, which is usually reduced to 4.22:1
To calculate the odds of making your flush by the river, You cannot just take the odds of making it in one card and multiply it by 2. (This logic would say that if 5 cards came out, you would be guaranteed of making your flush, which we know is not true.) You have to calculate the probability of not making you flush on either card. So, on the turn 38/47 cards won’t help and at the river 37/46 cards won’t help. The product of these probabilities is the chance of not making your flush, so:
(38/47)*(37/46)= 0.6503 or, you will miss 65% of the time.
Since we know how often you will miss, we can calculate how often you will make the flush by subtracting from 1.
1- 0.6503 = .3497 or 35%. The odds are .65: .35 or, dividing both sides by .35, we get 1.86:1

For an open ended straight draw we have 8 outs, so the math looks like this:
39/47=.8298 38/46= .8261 (.8298)*(.8261)= .6854
(1-.6854)= .3146 (31%) or 2.18:1 by the river and 4.88:1 on the turn.

To flop a flush with 2 suited cards we have to get 3 of our suit in 3 consecutive cards, so this time we must use the product of the probabilities of our suit coming on each card.
(9/50)*(8/49)*(7/48) = .0043 0.43% or 232.6:1

To flop a straight with 2 connecting cards 54-JT there will be 8 cards that connect with your 2, 8 that connect with those 3 then 8 that connect for the straight, so we get:
(8/50)*(8/49)*(8/48) = .0044 0.44% or 226.3:1

We can estimate the chances of you flopping a straight or a flush by adding the numbers together,
.44+.43 = .87% or 114:1
but that is not entirely accurate since some of your flush cards are also straight cards and vice versa.

To flop a straight flush to suited connectors, we have
(2/50)*(2/49)*(2/48) = 0.000068 or 14,705:1
And a royal flush to Aks (or any other 2 suited face cards) is 117,646:1

We can estimate the chances of flopping 4 to a flush by multiplying the chances of the first card being your suit by the chances of one of the other 2 cards being your suit.
(9/50)*[1- (41/49)*(40/48)]= 0.0558 or 5.58%
We then add this to the chances of the first card not being your suit, and the other 2 being your suit
(41/50)*(9/49)*(8/48) = 0.0251 or 2.51%
2.51 + 5.58 = 8.09%
To flop an open ended straight draw,
(8/50)*[1-(41/49)*(40/48)]= 0.0496 or 4.96%
(42/50)*(8/49)*(8/48)= 0.0229 or 2.29%
4.96+2.29= 7.25%

So you will flop one or the other about 15% of the time. Once you flop the draw, you will make it about a third of the time, so, adding the chance of flopping it, 6% of the time you start with suited connectors, you can expect to complete a straight or flush. (It is actually about 9% if you consider runner-runner)

[Demiparadigm]

Determining outs:

The number of outs you have is the number of cards that will give you the best hand, for example, when you have an open ended straight draw, 8 cards out of the 47 that you haven’t seen will give you the straight, but not necessarily the best hand.

When calculating outs, it's also important not to count the same outs twice. An example would be a flush draw in addition to an open ended straight draw. 9 cards will give you a flush, and 8 cards would give you a straight, but since 2 of the cards that would give you a straight would also give you a flush, you have 15 outs, not 17.

Also, sometimes an out for you really isn't a true out. An example would be chasing an open ended straight draw when two of another suit are on the table. In this regard, where you would normally have 8 total outs to hit your straight, 2 of those outs will result in three to a suit on the table. This makes a possible flush for your opponents. As a result, you really only have 6 outs for a nut straight draw. Another more complex example is when you are drawing to a straight that is not the nuts. If you hold 76s and the flop is 789, a 6 or a 10 would give you a straight, but the 10 also would make a straight for someone holding a J, giving you 4 outs, or you could be drawing dead against a player who is holding J,10.

This is why I recommend only drawing when you are drawing to the nuts.

[Demiparadigm]

Here is a table of your chances of improving your hand after the flop:

Outs River% Turn% River Odds:1 Turn Odds:1 Draw Type
2 8% 4% 12 22 Pocket Pair
3 13% 7% 7 14 Single Overcard
4 17% 9% 5 10 Inside Straight, Two Pair
5 20% 11% 4 8 One Pair
6 24% 13% 3.2 6.7 Two Overcards
7 28% 15% 2.6 5.6 3 of a kind
8 32% 17% 2.2 4.7 Open Ended Straight Draw
9 35% 19% 1.9 4.1 Flush Draw
10 42% 22% 1.6 3.6
11 42% 24% 1.4 3.2
12 45% 26% 1.2 2.8 Flush+Inside straight Draw
13 48% 28% 1.1 2.5
14 51% 30% 0.95 2.3
15 54% 33% 0.85 2.1 Flush+Open Ended Straight
16 57% 34% 0.75 1.9 Flush+OESD+One Overcard
17 60% 37% 0.66 1.7 Flush+OESD+Two Overcards

[TylerK]

Good post, but can you dumb it down a little?

I'm trying to figure out how I can determine the percentage of (for example) flopping trips.

I can figure out the odds of pairing one of the cards in my hand (3/50 + 3/49 + 3/48 right?), but I'm stuck from there.

[Demiparadigm]

um... not quite.

If you have a pocket pair (which is 17:1 odds)
you can determine the chances of one of your cards coming by multiplying the chances of it not coming,
So 2 cards will make your set, 48 won't
48/50 x 47/49 x 46/48 = 0.8824
we then subract this from 1 and get 0.1176
so the odds are 0.8824:0.1176
divide both sides by 0.1176 and we get
7.5:1

[TylerK]

No, not a set, trips...so two on the board to 1 in your hand.

[Demiparadigm]

If you have 2 different cards, to get trips we determine the chance of pairing one of our cards on the first card (6 cards will give us a pair) and multiply that to the chance of a card of the same value coming in the next 2 cards:
(6/50)*[1- (47/49)*(46/48)]= 0.00972
Add this to the chance of the first card not pairing one of our cards to the chance of the next card pairing one of our cards and the last filling the set.
44/50 x 6/49 x 2/48 = 0.00449
0.00449 + 0.00972 = 0.0142 or 1.4%

To convert this to odds, we subtract the probability of it occuring from 1 to get the probability of it not occuring (1- 0.0142 = 0.9858)
so our odds are 0.9858 : 0.0142 or 69:1

[Greedo017]

if the second card pairs your other hole card then you have twice the odds of the third card giving you trips, its just you'd have a full house but i think its worth mentioning in the trips category.

[koolmoe]

Just a tip: these problems are a bit easier if you just consider the number of boards that meet your requirements and divide that by the total number of boards. That way you don't get caught up in the "if the first card..., if the second card" thinking.

For example, if you have an unpaired hand, there are 2*(3*48) (that's 2*combin(3,2)*combin(48,1) - the two is for the two ways you have to hit trips) ways to make trips or better on the flop (3*2*44 if you want to rule out full houses and quads). There are (50*49*48)/(3*2*1) (that's combin(50,3)) possible flops.

So, the probability of flopping trips or better is 288/19600 = 0.0147

For those that don't know about computing combinations, they are taught in most introductory probability courses. You can probably find a book at your local library.

[koolmoe]

If you have 2 different cards, to get trips we determine the chance of pairing one of our cards on the first card (6 cards will give us a pair) and multiply that to the chance of a card of the same value coming in the next 2 cards:
(6/50)*[1- (47/49)*(46/48)]= 0.00972

As was pointed out, this calculation is slightly in error since it rules out the possibillity that the second and third flop cards pair the "other" hole card.

Note that it doesn't rule out flopping quads, though your next calculation does.

[Tottenham]

Is there an "easy on the eyes" chart of some kind on this info?

Calculating this on the fly in a game is near impossible, so I think straight memorization of some of the more common hands/situations might be a good idea.

[RiverMonkey]

How to calculate odds:
To calculate the odds of making your flush by the river, You cannot just take the odds of making it in one card and multiply it by 2. (This logic would say that if 5 cards came out, you would be guaranteed of making your flush, which we know is not true.)

Great post. I've read some of your other odds-related posts too and you clearly you have a keen mathematical mind and a knack for explaining card probabilities and odds etc. Your posts are very helpful and surely appreciated by all.

Now, while I completely agree with your numbers and methods in this post (at least the ones that I've looked at so far). I can't seem to convince myself that the specific reasoning mentioned in the one point above makes sense. Unless I'm misunderstanding what you intended to say, this sentence's reasoning cannot be accurate: "This logic would say that if 5 cards came out, you would be guaranteed of making your flush, which we know is not true",

Why? Well, if something is a statistical certainty (i.e. 'guaranteed'), and that something is realized through as a set of composite, independent events, then the probabilities for those individual events have to add up to 1. Which would imply that for your statement to be accurate the probability for hitting on the first card would have to be 50%. ???

By the way, I'm sure YOU know what I'm about to spew forth through and through and you don't need my clarification. I'm just adding this to the thread for the benefit of others who are not as mathematically inclined as you.

The reason you cannot just add the probabilities (which is what I assume you meant by "double") of the individual events together to arrive at the overall probability is that the events we are focusing on here are not independent of one another. i.e. hitting the flush on the river requires that it was not hit on the turn first. And, as we know, in order to calculate the overall probabilty of hitting "by the river" you need to first figure out the independent probabilities of each event (namely (i) hitting on turn, and (ii) hitting on river) and then add them together (since we do consider these events to be independent of one another)

Prob-River =

[Prob. of not hitting on Turn]*[Independent Prob. of hitting on River] =

[1-Prob.OnTurn]*[Prob.OnRiver]

As was already pointed out, Prob-"OnTurn" is straightforwardly calculated as: outs/47. Therefore [Prob. of not hitting on Turn] = 1 - outs/47 (since odds of not plus odds of yes must add to one).

Thus,

Prob. By River = [Ind. Prob. on River] + [Ind. Prob. on Turn]

= [1-outs/47]*[outs/46] + [outs/47]

Setting outs = 9 gives:

= 0.1581868 + 01.1914893
= 0.3496761

or about a 35% probability.

If you like working in 'odds against' better than '%-probabilities', then:

we have [100 - 34.96761]/[34.96761] to one

or about 1.86:1 against.

[JeffreyGB]

How to calculate odds:
To flop a flush with 2 suited cards we have to get 3 of our suit in 3 consecutive cards, so this time we must use the product of the probabilities of our suit coming on each card.
(9/50)*(8/49)*(7/48) = .0043 0.43% or 232.6:1

Just wanted to post a quick correction here, since I made the same mistake in another thread. There are actually 11 (then 10 then 9) flush cards.
11/50*10/49*9/48 = .0085 = 0.85% = 116.6:1

[HaD]

the correct probability to flop straight from connected cards is 1.32%

[HaD]

the correct probability to flop straight from connected cards is 1.32%

[Demiparadigm]

Figures someone would bring this back from the dead to point out I was wrong.
I agree Pyroxene's post is much more thorough than this one. I was just trying to do the math in my head.
cest la vie.

For answers to odds questions go here:
http://www.flopturnriver.com/phpBB2/...pic.php?t=9366

The notes on determining outs and chances of improving are in my opinion the most important and overlooked part of this thread. I am sure those are right, (the math is easier)