Realistic odds of hitting a flush after flopping four?

Here's my Math -- I'm looking for a little help.

In 9 handed play, there's 18 pocketed cards, so you would "expect" 4.5 of any one suit to be out there.

Given that the odds of the burn card being the suit are roughly .25, there's "9.5" cards of that suit remaining. So, if two come on the flop, there's roughly "7.5" cards left of that suit.

So, since you have 30 cards left, I assume the odds of catching a flush after a four-flop to be roughly 50%. Is this true?

Here's the hand that confuses me...

76,958,699 games 60.537 secs 1,271,267 games/sec

Board: Ac Ah Tc
Dead:

equity (%) win (%) tie (%)
Hand 1: 74.4460 % 74.45% 00.00% { As3s }
Hand 2: 25.5540 % 25.55% 00.00% { Kc7c }

The A3 has no draw, and can only be improved by another ace. And, even if make that ace a dead card, it really only moves the percentages a point or two. So -- given my math above how is this possible? H2 hits the flush and wins....but only has a 25% chance of doing so? Does this mean that the math just doesn't work out in real life? Or am I entering this incorrectly in poker stove?

Best,
EW

When you try to calculate odds the math is based on known cards and unknown card.
Lets say you hold 2 of a suite and the flop has 2 more of that suite. There are 9 remaining of that suite out of the 47 remaining card. The chance of completing the flush in the turn or river = 1-(38/47*37/46) = 0.35%.
In the example you gave there are less outs since 3c gives Hand1 a full house, and any cards that pairs the board gives Hand1 a full house, so it amount apprently to 25%

Originally Posted by TLR

When you try to calculate odds the math is based on known cards and unknown card.
Lets say you hold 2 of a suite and the flop has 2 more of that suite. There are 9 remaining of that suite out of the 47 remaining card. The chance of completing the flush in the turn or river = 1-(38/47*37/46) = 0.35%.
In the example you gave there are less outs since 3c gives Hand1 a full house, and any cards that pairs the board gives Hand1 a full house, so it amount apprently to 25%

Duh! Forgot about the boat. Now it makes sense.

Best,
EW

Part of the reason why getting all in with a flush or a straight draw against a set is real bad.

Yeah, a lot of people don't realize that a set on the turn improves more often to a boat or quads (10 outs) than a flush draw (9 outs) or OESD (8 outs).

Hm, a little side-note there. Say you have 22 and the flop is 2 5 8. Now you have 7 outs to a boat/quads. Then the turn is J. Now you have 10 outs to a boat/quads. So over turn+over that is an "average" of 8.5 outs, slightly less but obviously very comparable to a flush draw with 9 outs over turn+river.

Ofcourse you are 100% correct in what you said, but I give this perspective to avoid the confusion if someone thinks he can now count 10 outs over turn+river from his set.

Originally Posted by Xanadu

Yeah, a lot of people don't realize that a set on the turn improves more often to a boat or quads (10 outs) than a flush draw (9 outs) or OESD (8 outs).

If I am not mistaken a set is 30% to fill up with two cards left, a straight is 32% to complete, and a flush is 36% to complete.

Originally Posted by Renton

If I am not mistaken a set is 30% to fill up with two cards left, a straight is 32% to complete, and a flush is 36% to complete.

More accurate would be:

set 33%, straight 32%, flush 35%

Maybe you only accounted for a boat and not quads?

Originally Posted by jackvance

Originally Posted by Renton

If I am not mistaken a set is 30% to fill up with two cards left, a straight is 32% to complete, and a flush is 36% to complete.

More accurate would be:

set 33%, straight 32%, flush 35%

Maybe you only accounted for a boat and not quads?

Nope. Only remembered from a showdown of a set vs flush on tv. The percentage was probably twirked because TV accounts for dead dealt cards.

Originally Posted by Renton

Originally Posted by Xanadu

Yeah, a lot of people don't realize that a set on the turn improves more often to a boat or quads (10 outs) than a flush draw (9 outs) or OESD (8 outs).

If I am not mistaken a set is 30% to fill up with two cards left, a straight is 32% to complete, and a flush is 36% to complete.

I said on the turn ... read jackvance's post (looks like the clarification was needed jack)

Plus, if you add the quads, a set improves more often than a straight from the flop.

Originally Posted by Xanadu

Originally Posted by Renton

Originally Posted by Xanadu

Yeah, a lot of people don't realize that a set on the turn improves more often to a boat or quads (10 outs) than a flush draw (9 outs) or OESD (8 outs).

If I am not mistaken a set is 30% to fill up with two cards left, a straight is 32% to complete, and a flush is 36% to complete.

I said on the turn ... read jackvance's post (looks like the clarification was needed jack)

Plus, if you add the quads, a set improves more often than a straight from the flop.

my bad

st8-34.2%
boat/quads-35.5%
flush-36.5%

Str8 and flush are vs overpair
boat/quads is bottom set vs flopped straight

Exactness doesn't matter so much, all of them are slightly better than 2:1 most of the time.

An idea of how it can change:
OESD vs top two is only 30%, because the top two has 4 outs to make a boat.

Originally Posted by NoSocksLLC

Given that the odds of the burn card being the suit are roughly .25, there's "9.5" cards of that suit remaining.

dont calculate for the burn card...you dont know what it is so it might as well being put back on the bottem of the deck. There is still at 1/47 chance of each card coming on the turn (2 cards in your hand and 3 on the flop cant come again so it is 47 not 52)

A lot of people mistakenly try to factor in burn cards and unseen cards when they compute odds. While it is wrong to do so, it can be difficult to explain to people why that is so.

I think I have a way of explaining it that is easy to follow. If it clicks for you, great. If not, oh well.

Assume a heads up, 2 man match without burn cards. You are dealt the 1st and 3rd card, your opponent is dealt the 2nd and 4th card. Cards 5, 6 and 7 form the flop. You have four to a diamond flush. You will hit your flush if, when the deck was shuffled, a diamond landed on either of the 8th or 9th card.

Assume the same situation but there is a PF-burn, pre-turn and pre-river burn. Now cards 6, 7 and 8 form the flop, 10 is the turn and 12 is the river. You will hit your flush if, when the deck was shuffled, a diamond landed on the 10th or 12th card.

Assume you are at a 5 handed table with no burn cards. Cards 11, 12 and 13 form the flop, 14 the turn, 15 the river. You will hit your flush if, when the deck was shuffled, a diamond landed on the 14th or 15th card.

And so it goes, no matter how many people are in the hand, no matter if you burn or not or even how many cards you choose to burn, you will hit your flush if a diamond landed in either of 2 predetermined slots in the deck when it was shuffled; with the only other known piece of information being 4 diamonds and one non-diamond are ineligible because they are in your hand or in the flop.

This is one of the things that makes Hold'em so much easier than a game like 7 card stud. In Hold'em, you know your two cards, then you know three community cards, then you know a community turn and then you know a community river. It makes the odds trivially easy to compute. This is why you can get sheets for Hold'em that will tell you the exact odds of improving to certain hands given you hole cards, then given the flop, then given the turn.

But in 7 card stud, you are constantly learning about entire sets of cards and none of them are community. So the odds of you improving are sliding all over the place based on 2-8 cards getting shown on each of 4 streets. And, as they are not community cards, the odds of your opponent's hands improving are fluctuating wildly. Assume an opponent has four to a flush showing. If you have none of that suit and none of the other 0-20 other face up cards are of that suit, you have to assume that it is very possible that he ahs a flush. However, if between your cards and the other 0-20 face up cards you can acocunt for 8 of the other cards of that suit, then it is significantly less likely.

That is why there are no equivalent sheets of odds for 7 card stud. The odds vary enormously from hand to hand and street to street.