I'm not sure where else to ask this question, if there was an l/c thread I would have asked it there. Sorry in advance for the lame thread.

I'm on a super cold run of cards primarily with how frequently I'm flopping pairs. I think there's a 28.6% chance to flop a pair, I estimate for the past 50,000 hands or so I'm flopping a pair at about a rate of 1 for every 10 flops, maybe lower. I haven't been tracking this so I could be unintentionally exaggerating my bad luck but I feel this is close to correct. What are the odds of this happening over a 50,000 hand sample (using the rate of 1 in 10)?

Yeah I am like super eager to do the math on something that you think might have happened to you but you haven't actually even bothered to check

I don't have any poker software installed on my computer other than the PokerStars client and I don't know how to do the math for this, how am I supposed to figure out the likelihood without asking? If making this thread was an infraction just lock it and I'll figure it out with Google.

Flopping a pair only 1 in 10 flops over a large sample is essentially impossible. Your recollection is wrong. Request your hand histories and count them instead of guessing.

And the number you need is not total hands. It's only hands where you saw a flop while holding unpaired cards.

You'll get around 32% including trips and 2 pairs.

You're right. I don't know why that didn't cross my mind when I was thinking about this. I've actually kept on the fly records over a reasonably decent sample of HUSNG play, where you see nearly every flop, and had days where I've played 500-750 hands and gone through 20 hand streaks with only a couple of pairs for pretty much the whole session. Many of these hands are going to show down so I definitely felt like some statistical outlier missing so many boards. Of course it sounds like I just don't understand variance but when this happens over hundreds of games you start to feel it. You also have people ready to jump down your throat because they think you're whining, when you're just running bad, and only talking about it because you're morbidly curious about how unlikely it was to happen.

As others have pointed out, the odds of what you are describing is virtually nil, so it is much more likely that what you are describing did not actually occur.

Binomial probability calculators exist in many forms. Many calculators and software programs (such as Excel) have them built in. Or you can search the internet for the formulas.

I'd suggest following the suggestion of the post above this one if you'd like to pursue this further.

If you have any further questions on this, feel free to post them here.

I'm not suggesting you should figure out the likelihood, I'm suggesting that if you're going to be concerned about the chance of "something" happening, you should at least know what that "something" IS. That is, you need to know how many hands are in your sample and how many meet your criteria. Without that it's pretty pointless.

It's not perfectly clear to me what you want.

If you're talking about Texas hold 'em and you hold, for example, AK, there are 3 missing aces, 3 missing kings, and 44 missing cards which are neither aces nor kings.

In a three card flop where we are not interested in suits, and not interested in permutations, here are all the possible combinations:
AAA,
AAK,
AKK,
KKK,
AAN,
AKN,
KKN,
ANN,
KNN, and
NNN.

And here are the possible ways to make each of the above combinations:
1---AAA,
9---AAK,
9---AKK,
1---KKK
132---AAN,
396---AKN,
132---KKN,
2838---ANN,
2838---KNN, and
13244---NNN.

That should all add up to C(50,3)= 19600. Does it?
1+9+9+1+132+396+132+2838+2838+13244=19600.
(That was our check).

Since you hold AK, if you want just flops that have an ace or king, in order for you to flop a pair of aces or a pair of kings, there are 5676 of those flops.

Finally 5676/19600= about 29%.

(Or if you count flops that have exactly a pair of something other than ace or king and no aces or kings as flopping a pair, we'd get a different number).

At the rate of 1 in 10, the odds of not flopping a pair would be 9 to 1.

But as you see from above, when you hold AK, the odds of not flopping 1 ace or 1 king are 13924 to 5676 or about 2.453 to 1.

Buzz

Usually we would count everything except flops with no A or K, meaning trips, full houses, 2 pairs and quads would still count as flopping a paired hole card. He may have meant "exactly" one pair, but that isn't as useful a statistic.

So that figure would be 1 - (13244/19600) = 32.4%

Now he has both.

Assume his 1 in 10 numbers are correct.
Could this be explained by variance?
An extreme outlier?
Or would it be a glitch in the poker program?

The probability of flopping a pair is 0.2896 given two unsuited cards. The probability that you will flop 5000 or fewer pairs in a sample of 50000 flops is as close to zero as you can get.