# Statistical analysis of over 3000 cards dealt by Replay

As a fun experiment, I’ve been tracking my hole cards for the past 1700+ hands, and applied some spreadsheet wizardry to see what I can see.

For the most part, we’re pretty close to what probability says should be expected.

Stats Count Actual Expected
hands 1727 100% 100%
suited 393 22.8% 24%
offsuited 1334 77.2% 76%
offsuited exluding pairs 1238 71.7% 71%
pocket pairs 96 5.6% 6%
connectors 237 13.7% 15.69%
gappers 230 13.3%
disconnected 798 46.2%
rags 611 35.4% 37%
suited aces 62 3.6% 1.81%
suited kings 61 3.5% 1.81%
suited queens 43 2.5% 1.81%
weak faces 644 37.3% 18.10%
suited weak faces 140 8.1% 4.52%
AA 6 0.4% 0.45%
KK 10 0.7% 0.45%
QQ 6 0.4% 0.45%
JJ 4 0.3% 0.45%
TT 8 0.6% 0.45%
99 5 0.4% 0.45%
88 10 0.7% 0.45%
77 7 0.5% 0.45%
66 14 1.0% 0.45%
55 7 0.5% 0.45%
44 5 0.4% 0.45%
33 5 0.4% 0.45%
22 9 0.7% 0.45%
AKs 4 0.3% 0.15%
AKo 19 1.4% 0.45%
AK-all 23 1.7% 0.60%
Mean Median Mode
Top card Rank 10.3 11 14
Bottom card Rank 5.8 5 2

Distribution of card ranks:

The expected distribution should be flat, and if it were flat, we’d see 266 occurrences of each rank, but as we see here, there’s quite a lot of variance evident. I’m not sure how many cards I’d need to draw to see these bars smooth out, assuming a fair RNG.

I took the time to calculate the standard deviation for this distribution, which is 29. It’s been a long time since I studied math, and I never did a lot with statistics, so I’m not clear what this tells me, but with this distribution, we see 5 card ranks that are outside of 1 standard deviation of the average of 266 occurrences per rank:

• Deuce (+35),
• Five (-62),
• Six (-39),
• Jack (+29), and
• Queen (-43).
Card Rank Count
A 277 8.02% 11
K 280 8.11% 14
Q 223 6.46% -43
J 295 8.56% 29
T 286 8.30% 20
9 269 7.81% 3
8 287 8.34% 21
7 280 8.14% 14
6 227 6.60% -39
5 204 5.94% -62
4 257 7.48% -9
3 268 7.81% 2
2 301 8.78% 35
Average count 266
Standard deviation (σ) +/-29
1-σ Range 237 294

If I keep tracking this, it’ll be interesting to see what the numbers look like after I have 10x the data that I’ve gathered so far; I’d expect that as I enter more hands, the variance we see in the distribution of hands should smooth out more. I’m a bit surprised at the amount of variation that we see after 1700+ hands, but I’m not sure if this is truly outside of what probability tells us we should expect from a fair deck. But do I think my hole cards have been reasonably fair over this sample.

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Interesting, but as you said, still a small sample size.

In a normal distribution, 50% of the values should be above the mean, and 50% below.

68% of the values should fall within +/- 1 SD of the mean.

95% of the values will be within +/- 2 SDs of the mean.

97.5% should land within +/- 3 SDs of the mean.

Good to see you collecting actual data, keep it up!

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small sample

I agree that the sample size is still on the small side; I’d like to get an idea how to calculate what sample size I would need in order to expect to see the distribution of the cards match closely with what probability expects.

I’m sure it has to do with the fact that there are 13 ranks. With coin flips, there’s only two possible outcomes, and you see actuals meet expected pretty quickly with a fair coin. I’m not sure how many flips it takes to feel confident that a coin is fair, either, but as a rough guess I’d say it’s reasonably clear after 100 trials, and after 1000 or 10000 trials, confidence should be absolutely solid.

With a 13-sided coin, would you need 13 times as many trials as you’d need for a 2-sided coin? ^13 as many trials? 13! as many trials? I’m not sure how to apply mathematical principles to inform me how to know.

For that matter, I could use some help double checking some of my calculations for the expected percentage of “weak faces” (defined as high card 10+, low card <8).

I calculated it as there are 20 cards in the deck with a rank of 10+, and 24 cards in the deck with a rank of 2-7, so 20/52*24/51 = 18.10%, but my actual observed percentage of weak face cards is 37%, which seems aberrant if my expectation is correct.

But actual % is so far off expected % that I suspect that my expectation isn’t. Likewise with suited weak faces. (20/52 * 6/51).

I’m not really sure how useful it is to know how often I get dealt weak face cards, anyway. It might be more useful to know how often I get dealt suited weak Aces and Kings only.

The 18% seems right, but that’s the % of hands out of your total population that should be weak face cards. So, out of your 1,727 hands, 311 of them should have been weak face cards.

Once the first card is in the range of 10-A, you should see the second card be in the 2-7 range 47% of the time.

If your first card is 2-7, the second will be 10-A (20/51) * 100 = 39%.

Make sure the numbers you calculate are answering the actual questions you ask!

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What I’d like to see added is the winning hand percentage of all those hands. especialy AA and KK.
And also the ‘hindsight winning hand’ with a say 6 & 9 hand and the table produces a 7-8-10.
Also the sequence of hands would be interesting. Do you get AA and the next hand KK or 6-9? What is the frequency of winning hands?

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Playstation 3 WSOP had a feature like this. I think it broke down win/loss for all specific hole cards dealt. Its was interesting although it was against AI opponents. It would be useful IMO if RP could provide a similarish feature. OFC your AK etc win/loss could be bad bc its played badly etc. Still interesting & easily available data could support RNG & fairness etc.

I have heard other cash sites, staff & players talking about providing statistical data for each player etc in a bid to strengthen players trust the site is fair & RNG is legit etc. Im not sceptical nor do want better proof, but the data would be useful & interesting.

I would be interested in such data too. I win a ship ton with AA & do ok with AK, but feel like RP is out to get me when im dealt KK. Feels like every time I get KK the flop has Ace nemesis. So tilting.

Is this a question? Maybe you meant I do & not do I? There is no question mark. OFC its a small sample.

Consideration of whether its “fair deck” or fair deal the cards on the flop, turn river are presumably important? Seems like your hole cards are standard? Im not sure what you expect to learn from this, but maybe reinforce the holecards are RNG & fair.

My Achilles heal is 10-10.
I seldom win with that combo, even with aggressive bidding.

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I hate JJ more. It loses me more chips because I’m more likely to play them stronger–and longer.

Right, not a question, other than were my calculations for the expected percentages correct.

I mainly did this to see how “fair” (by which I mean randomly distributed) the hole cards were dealt to me, and overall this sample has shown that they do appear to be reasonably random.

There’s plenty of other ways a game could be rigged, of course, that this exercise is not looking for, and could not detect. Im looking to not go into those topics in this thread, though.

For this exercise I’ve been recording only the hole cards, nothing else, so I do not have outcomes. If Replay ever makes it possible to download my entire hand history as data, I’ll surely do some extensive analysis of it.

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At least with TT you can get many over cards that will force you to slow down. I play LOW & MED stakes and the scenarios are vastly different and completely unpredictable.

I’ve raised KK on 9 player tables LOW stakes with 4 or 5 callers with an A flop what are the chances someone hits? I would think pretty high but I cBet anyway & they call down with nothing.

High stakes 6MAX raising KK isolated to 1-2 players and OFC they have Ace rags & hit an Ace. Super frustrating.

Generally I play AA & KK the same but I rarely lose with AA. KK is my Achilles heel.

An interesting experiment would be to reproduce a comparable dataset using your own randomly dealt cards. Do it 1000 times (or more). Check the standard deviation of each one. Does your replay standard deviation look like an outlier?

Are there stats on flops and in particular suited flops. Seems these a far more common here as are flushes. Not a math guy so maybe you can set me straight Pug (no pun intended)

Here you go @Itapua0309. This is from a couple thousand hands on Replay. Nothing out of line with what you should expect.

I haven’t been tracking board cards, just hole cards.

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How difficult is it to track and how do you do it?

Do you have a spreadsheet open and enter data or manually write it down etc?

Others have suggested it would be good if there was software to easily gather data, but I dont think there is any? Savvy Poker Players have created software to improve poker skills and give advantageous data like HUD, range graphs (& more) plus even Solvers which basically all cost money. I would guess these tools are created by skilled, savvy, technical IT, mathematically minded poker players.

I guess in general there are not enough good players interested in creating this software and also there isn’t a recognisable demand to create and sell the software?

It would be interesting for a large amount of data to become available to the public for various poker sites, including free poker IMO.

I don’t want to play cyborgs, or become one; I want to be good at poker.

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I should say, though, that I do study the game away from the table using computer aids and tools, and I think that’s fine, it helps me see things that are not easy to see unaided, and helps me to understand the math underlying the game, and become a better player.

Math major here specializing in stats. I’m not going to do much work on this because I don’t have to, but the distribution seems very fair, about what you’d expect. Given a larger sample size, I’d expect tighter trend lines, especially for the 13 cards seen, but for 3000 and the given SD it is very reasonable. I’d love to see a follow up.