The Pair vs Pair Paradox

One of the most frustrating things in Hold 'Em is getting a good pocket pair, seeing all lower cards come on the board, and still losing at showdown to a higher pocket pair or a lower pocket pair that hit a set.

After a while it can seem odd. You don't get pairs that often, and when you do it seems that other players have pairs more often than what "feels right."

The odds you'll be dealt a pocket pair in Hold 'Em is once out of every 17 hands, or 5.8%. So you don't seem them that often.

Because you only see a pocket pair 5.8% of the time in your own hand, the mind naturally assumes since this is an infrequent event, you will not be likely to face another pair. Plus with any ten handed Hold 'Em hand, the odds that more than one player is dealt a pocket pair is 11.4%.

So, the odds of you getting a pocket pair is 5.8% and the odds of more than one pocket pair on the table is only 11.4%. So why do you see opponents with pairs so often?

The simple reason is that it happens more often than you think it should. Once you receive your cards and see a pocket pair, the odds another player has a pocket pair are 42.5%.

You read that right. The odds of more than one player on a ten-handed Hold 'Em table receiving a pocket pair is 11.4%. But when YOU see your pocket pair, the odds of another player holding one is 42.5%

This is what we call the Pair vs. Pair Paradox. It’s not really a paradox, this is just how numbers work. And when these things happen, some players begin to suspect foul play or that the site is dealing rigged hands.

Lots of players who worry about a site dealing rigged hands typically complain that a site "doesn't feel right." As we've discussed, the evidence coming from people worried about poker sites cheating by scripting how the cards are dealt is scant. Most of the concern is based on "hunches." And "paradoxes" like this create those "hunches" and "feelings."

But the "feeling" that opponents too often have pairs the same time you do is wrong -- unless you "feel" that it should happen about 42% of the time -- typically far above what many would estimate.

The REASON for this is that once you look down at your pocket pair, we've already reached the first step in the probability equation that showed the 11.4% chance more than one player have pocket pairs. Once you see your pair, there is now a 100% chance that YOU have a pocket pair. And now there are nine other players who each individually have an approximate 5.8% of having one.

Think about it this way:

What are the chances of you flipping a coin HEADS three times in a row?

Simple answer: 50% * 50% * 50% = 12.5%

What are the chances of you flipping a coin HEADS three times in a row after you've already flipped HEADS twice in a row?

Simple answer: 100% * 100% * 50% = 50%

The bottom line: When you have a pocket pair in Hold 'Em, the chances another player on your 10-player table also has a pocket pair is over 42%. When you see those hands, don't think the site is rigging the deal to build pots. Think that this is how things work in a random system.

Why is this a paradox? It's just the probability of an event occurring given that another event has occurred. Seems like simple statistics to me.

If this is the common conception, this is a good informative post

Play it just like any other hand. Watch the other players and their raises. If you flop a set then bet it. I cracked aces with pocket 3s in our live game tonight. If you have a small pair and feel you're beat then you gotta give it up. I'm not sure about a paradox, maybe you are just thinking too much into it.

Its not a paradox, and its not independent property, as you seem to assume.
In the coinflip example its a binominal distribution, but in the pair example it as hypogeometrical distribution, so you need to take that into calculation.
So calculating the chance that any of your nine opponents have a pair differs from calculating the chance that any of your nine opponents have a pair provided you have one.
In the first example: P(x=anyone of the nine players has a pair)=42.05%
In the second example: P(x=anyone of the nine players has a pair|you have a pair)=40.86%

Interesting post.

I have to admit, I do sometimes feel a bit cheated when I get QQ and someone has either KK or AA.

It is very hard to handle, but it is understandable why people feel aggrieved, especially if they were card dead for a long period before getting a strong hand.

[ ] paradox
[x] pretty good thread