Luck Bell Curve

[Joshann]

I love your luck bell curve for cash games. So I am trying to do something similar for tournaments except that I'm just making reports -- I don't know how to do a bell curve. Anyway, could you tell me what numbers were used in the Luck Bell Curve? In other words, what are the numbers that are considered Extremely Unlucky, Very Unlucky, Unlucky, Normal, Lucky, Very Lucky, and Extremely Lucky for flushes, for straights, and for sets?

Thanks!

[kraada]

It's based on standard deviations away from the mean. The expected value is considered normal and each color break is plus (or minus) one standard deviation. Anything above or below 3SD is just graphed at the extremes (which should be very rare as sample sizes grow).

[Joshann]

Right, I know in general how a bell curve works, but I guess I'm not understanding your bell curve. I thought I did, but I guess not. I thought you could give me the actual values for the mean, and each standard deviation away from the mean. I thought that the mean would be a fixed number based on statistical probabilities. For example, I was guessing that you know that in general, a person should flop a set when they hold a pocket pair X percent of the time, so the mean (normal) for sets in your bell curve would always be X (I think it's 11.8, isn't it?). Then "Lucky" would be X + 1 standard deviation or Y. Isn't Y always the same for sets? You can't give me that number? Do you see what I'm saying?

[kraada]

The percentage for flopping a set or better is always fixed (at 10.5%) - we do use sets or better for this report - but the actual standard deviation changes based on the sample size.

The way this works is that first we find the standard deviation of the probability in question using the formula:

sqrt(p * (1-p)^2 + (1-p) * p ^2)

and then you multiply that by the square root of the sample size. So plugging in .105 into that formula only gives you half of the equation and the other half varies for every person. You can see your own sample size though in our log file - just start PT4 with the Logging Enabled link in your start menu, load up the report, and then click File -> Show User Data Folder and open the text file called PokerTracker in that folder. You'll find the values you want at the bottom of the file and it'll look something like this:

2012/11/03 10:50:48.655 [00088]: [00000990][ptQueryCtrlBase::Populate] Thread [0x00000990]: Begin
2012/11/03 10:50:48.655 [00000]: [00000990][ptQueryBellCurveGraph] Flushes weighted ev: -1.49
2012/11/03 10:50:48.655 [00000]: [00000990][ptQueryBellCurveGraph] Flushes: Flop: 27 / 178 (0.00); Turn: 46 / 262 (0.00)
2012/11/03 10:50:48.655 [00000]: [00000990][ptQueryBellCurveGraph] Straights weighted ev: 0.85
2012/11/03 10:50:48.655 [00001]: [00000990][ptQueryBellCurveGraph] Straights: Flop: 18 / 113 (0.00); Turn: 23 / 127 (0.00); GS Flop: 33 / 336 (0.00); Turn: 39 / 402 (0.00)
2012/11/03 10:50:48.655 [00000]: [00000990][ptQueryBellCurveGraph] Sets ev: 0.41
2012/11/03 10:50:48.655 [00000]: [00000990][ptQueryBellCurveGraph] Sets: Pocket Pairs: 1949; Sets: 235

This is from my graph where flushes are shown at -1.49 (halfway between unlucky and very unlucky), straight at +.85 (almost lucky) and sets at +.41 (a little under halfway to lucky). And you can see how often every individual draw was hit as the results are all listed there.

[Joshann]

I see. The actual value of each standard deviation varies based on the sample size. OK, I got that part. So the percentage for flopping a set or better (the mean for sets) is 10.5%. What is the mean for straights? What is the mean for flushes?

Thanks for all your help!

[kraada]

Since variance is additive you can do:

p * (1-p)^2 + (1-p) * p ^2

based on the number of outs for each thing you want, then sum them, then take the square root of that number.

So for flushes, you use the probabilities of each draw hitting - 9/47 and 8/46, run each through the above formula, add the results together and take the square root.

For straights you want to do that for OESDs and gutshots and then add the resulting 4 numbers together before taking the square root.

[Joshann]

Now I'm completely confused. Never mind my last question. How about this? What is the probability (percentage) that a person will hit a straight by the river if they flopped a straight draw? What is the probability that a person will hit a flush by the river if they flopped a flush draw?

[kraada]

For a flush draw you take your number of outs and divide by the number of cards unknown.

On the flop you have 9 (13 - 4) cards that help your flush. There are 47 (52 - 3 flop cards - 2 hole cards) cards that could be dealt. So you're 9/47 to hit your flush on the turn. If you miss your flush on the turn there are still 9 cards that help but 46 unknown cards - so 9/46 to hit on the river.

To combine that all 9/47 + (38/47 * 9/46) = 34.967%.

But this assumes you saw both the turn and the river - our luck graph does not assume that. If you saw the turn and there was no river we still take that hand into account - which is why we use the method we do.

A complete list of odds for completion of various draws is available here (among other places, google just dropped me there first), but I thought you might like to know how these numbers were achieved not just what they were

You can figure out the straight draw options by looking for either 4 or 8 outs depending on the type of straight draw you've got.

[Joshann]

Thanks so much for all your help!!!