formula for calculating equity for hands more than 8 outs on the turn

[Walker]

It was in one of the old issues of 2P2 Internet Mag. Anybody know what it is?

[baumer]

if X = number of outs
2 cards in your hand
4 cards on the board
52 - 6 = 46 cards left in the deck

if you are certain you are behind, and also certain if you hit one of your X outs you will win...

Equity = X / 46

eg. for X = 9,

your equity will be 9/46 = 0.196, or 19.6%

[Walker]

Thanks dude, but I was hoping for something more simplified that I could do in my head on the fly. I normally multiply outs X4 with two cards to come and X2 with just one card to come. This formula works good with a small number of outs, but over eight or so there is some error.

Edit: With that formula and two cards to come, there's less then 1.5% error with 8 outs and only 5.7% with 21 outs. I guess I can live with that.

you should add 1 at the end of your calculations.


number of outs, say 8, times 2 which is 16, and then I add 1 to get a percentage. thus 17% chance of hitting on the next card. actual 8/47 = 17.02%, 8/46 = 17.39%.

heres another couple examples: 12 outs: 12*2=24 24+1 = 25% Actual: 12/47 = 25.5%, 12/46 = 26.1%.

for 2 outs the estimate says 5%. actual is 4.2% and 4.3% for turn and river respectively.

i guess this formula works best when you have around 8 outs.

Outs:Fraction of 990
1:44
2:87
3:129
4:170
5:210
6:249
7:287
8:324
9:360

Now, since these are out of 990, you can add 1/100 to make up the thousandth's.
For example: 8 outs is 324 outs .324+.003 => .327
(This is usually a very small correction.)
This error is quite small (less than, for example the board's chance to make a runner runner 3 of a kind on an unpaired flop) and is not likely to be worth more than a small fraction of a BB/100 hands over time.

Unless you're going to effectively consider all possible rivers and turns, you're not operating on a sufficiently accurate scale to care about anything but .04 per out.

Figuring .02 per out with one card to come is a significantly less accurate estimate than .04 per out with two cards to come. The actual value is .0227...
So you can get a large relative improvement by figuring .0225 (2 1/4%) per out. Fractions of 44 are also sufficiently easy to handle that you can do them in your head. However, once again, I expect that this type of improvement is worth very little in terms of BB/100 hands in the long run.

In summation:
Both .04 per out with 2 to come, and .02 per out with 1 to come are sufficiently accurate that your efforts at improvement are likely to be better used elsewhere.

[MyTurn2Raise]

[ QUOTE ]
Thanks dude, but I was hoping for something more simplified that I could do in my head on the fly. I normally multiply outs X4 with two cards to come and X2 with just one card to come. This formula works good with a small number of outs, but over eight or so there is some error.

Edit: With that formula and two cards to come, there's less then 1.5% error with 8 outs and only 5.7% with 21 outs. I guess I can live with that.

[/ QUOTE ]

With one card to come, multiply the number of outs by 2.17391304357 or 2.2 to be close (like kilograms to pounds).

Unless you can do combinations in your head, two cards to come are difficult. However, here is the shortcut I used when learning. For less than 10 outs, just multiply by 4. For 10 and more outs, multiply by 4 and subtract the value of the second digit plus 1. For example, with 15 outs, multiply 15 X 4 , then subtract 6 (the 5 from the second digit of 15 plus1). This gives 54--the actual chances are 54.1%. For 20 and up, you're screwed. (Hope you dont play omaha8)

With two cards
outs/ my method/ actual odds
1/4/4.3
2/8/8.4
3/12/12.5
4/16/16.5
5/20/20.4
6/24/24.1
7/28/27.8
8/32/31.5
9/36/35
10/39/38.4
11/42/41.7
12/45/45
13/48/48.1
14/51/51.2
15/54/54.1
16/57/57
17/60/59.8
18/63/62.4
19/66/65
20/69/67.5

Oh, yeah, are we playing Hold-Em or Omaha (or Stud which is even more involved).

[Guernica4000]

This is pretty simple:

Take the flush draw on the flop 4x9= 36%

For every number of outs over 8 take that number and subtract it from the number of total outs 9-8= 1 36%-1 = 35% and you get the exact number.

Here it is for 15 outs. 15 x 4= 60%

15-8 = 7 60% -7 = 53%

Your 15 outs are 53% favorite.

[dfan]

And you can make it even easier, since that formula simplifies to:

Let Y = # of outs
Then
4Y - (Y - 8) =
= 4Y - Y + 8
= 3Y + 8

So 15 outs = (3 * 15) + 8 = 53%
9 outs = (3 * 9) + 8 = 35%

Quite accurate for 9 outs through 20 outs.
For 1 to 8 outs just use 4*Y.