# Looking for mathematical paradoxes in Poker

I have been working on mathematical paradoxes.

I am looking for paradoxical situations in Poker for a deep study. I found things, like Morton's Theorem, Implicit Collusion or "all in" strategy, could you help me to find more?

• I'm not certain this type of question is a good fit for this site, although it is interesting! It could be a good jumping off point for generating some extra questions on poker.se. For that reason I hope you get some good answers ;) – Toby Booth Oct 29 '15 at 0:20
• A paradox is when one uses generally acceptable logic to, seemingly, soundly conclude something that is prima-facie illogical. Perhaps apply Zeno's paradox to pushing all in? Read out an equation that incorrectly subtracted, or divided as a story. Here is an example: en.wikipedia.org/wiki/Missing_dollar_riddle – Snipe Dec 17 '15 at 18:19
• Probably a little bit late: Do situations where ICM advises you to fold aces in Holdem qualify? – Drunix Jul 19 '16 at 6:12
• Would you make a distinction between a paradox and something which is just highly counter-intuitive? – 3N1GM4 Jan 5 '17 at 15:11

Not sure if this is really a paradox, but...

In a heads-up game your opponent truthfully confides in you that he is playing a 100% range.

So you think to yourself, "Great, from now on I will play a 70% range and I will be profitable because I’m ahead of his range."

You are dealt a new hand and see that it is at the bottom of your range, but you no-longer want to play the hand because it is behind most of your opponents range.

• Not seeing how 69% is behind a random hand (50%). – paparazzo May 4 '18 at 14:13
• @paparazzo because 69 > 50. Isn't that the definition of being behind? – Buh Buh May 4 '18 at 14:25
• Sure not my definition. – paparazzo May 4 '18 at 14:27
• I think we might be looking at it from opposite ends. I meant 69th percentile as in a low quality hand. Not a high quality. Perhaps I should have written it as 31st percentile? Does that help? – Buh Buh May 4 '18 at 14:35
• Playing the bottom of a 70% range is worse than playing the average of a 100% range – Buh Buh May 4 '18 at 14:36

Not sure if this is a paradox as much as counter-intuitive, but according to Phil Gordon, a rare case of over-cards being a slight favorite to a pocket pair is JTs vs. 55.

• That's just one example of the not-really-a-paradox of the non-transitive value of starting hands. Another is 22 > AK > JTs > 22. – Lee Daniel Crocker Jun 9 '17 at 2:11
• @LeeDanielCrocker :o could you send me a link regarding that? Sounds interesting. – Luis Masuelli Apr 5 '18 at 15:54
• (haha sorry for the late! this question was bumped and I read it just now) – Luis Masuelli Apr 5 '18 at 15:55

Ace high straight is highest but it is actually easier to make because there are no blockers above.

To me this is a Best hand versus AA as every way I look at it it seems like 67s should be better.

How almost all hands have at least a 33 percent chance of winning in a showdown against a pre-flop raiser. But yet they rarely actually win 33 percent of the time.