Calculating the odds of making a draw on turn or at river

[holdemsucks]

This question uses Limit Texas hold'em as its game.
I've been reading Small Stakes Hold'em and more than anything else I'm enjoying a lot of the maths!
Now here's what I'm curious about: if I have a gutshot straight draw on the turn then I have seen 6 cards and there are 4 cards (just assume that I can only win if I make the straight) that will make the nut straight. So my odds are 46 to 4 or 11.5 to 1. However, the authors say that the 'real break-even odds' are one less: 10.5 to 1. This isn't the question relating to the subject, but why do you need to subtract 1?

But now for the main question. In small stakes hold'em it only explains the mathematical formula for a draw on the turn to be made on the river. How about on the flop, with 2 cards to come, where the draw either completes on the turn or the river? My library doesn't have Hold'em for advanced players. [img]/images/graemlins/frown.gif[/img] I can't seem to find 'the formula' on the internet.

[Yepitis]

Someone else can explain this better I'm sure but the 10.5 to 1 is simply the way it is written out, 1 time you make your draw 10.5 you don't.

To figure this for the turn and river you can just divid by your two chances. So the 10.5 to 1 is about 5.25 to 1. In other words with two cards to come you a getting 8 cahnces in 47 to hit your gutshot. You get a little bit better odds knowing one more card on the turn but for all practial purposes this works.

[RobNottsUk]

[ QUOTE ]
This question uses Limit Texas hold'em as its game.
I've been reading Small Stakes Hold'em and more than anything else I'm enjoying a lot of the maths!
Now here's what I'm curious about: if I have a gutshot straight draw on the turn then I have seen 6 cards and there are 4 cards (just assume that I can only win if I make the straight) that will make the nut straight. So my odds are 46 to 4 or 11.5 to 1. However, the authors say that the 'real break-even odds' are one less: 10.5 to 1. This isn't the question relating to the subject, but why do you need to subtract 1?


[/ QUOTE ]

4 cards are hits, 42 cards misses out of 46 cards.
4 / 46 = 1 / 11.5 or in gamblers odds 4 - 42 10.5 to 1

I think, the real break even odds are slightly less because of implied odds of draws, where you expect to earn some additional bets on the river when you hit.

In this case,

10.5 times you spend a bet and earn nothing
1 time you make nut str8 for 1 bet, and win pot + all bets on the river (assuming no split pot)

When SSHE talks about "true break even" odds, I think it´s taking into account future betting. But if you post the exact page in SSHE of your example, the context will be clearer so any explanation can be more accurate.

[PantsOnFire]

[ QUOTE ]
So my odds are 46 to 4 or 11.5 to 1.

[/ QUOTE ]
No, your odds are 42 to 4. There are 46 cards, 42 are bad 4 are good. Your percentage (which is also your pot equity) is 4 out of 46 or 8.7%.

Think of it this way. Odds against are the ratio of bad outcomes to good outcomes. Percentage for is the ratio of good outcomes to all possible outcomes.

[phiphika1453]

quick math:

outs on flop x 4 = rough estimate
outs on turn x 2 + 2 = rough esitmate

try these out and you will see they are close enough for government work.