The Poker Paradox 

What if you don't consider the ranking when counting ways to get a hand? Like, you get a hand with the joker, which can be turned into either two pair or full house. Add +1 to the counter of both. Of course, you would always choose full house, since it is higher in the list, but that does not matter, you still theoretically can get two pair. That way we will get unambiguous number for "ways to get a hand", and we can rank hands by that number afterwards

The root of the paradox is that you have two things whose definiton depends to the other. The rankings determine the probability, but the probability determine the rankings. It's the same as the paradox where you say "the next sentence is true" and "the previous sentence is false". Most paradoxes in math are from things like this where there is a "loop", like self-referrential objects, e.g in Russell's paradox. So we always avoid them.

This video must be just a by-product of Zach's preparation for his next high-stakes poker game against James Bond

I mean, the resolution is obvious, right? You just count how many hands could be treated as the given rank. So a hand with a single pair and a joker should count as both a two pair and also a three of a kind, since you could treat it as either type of hand. Instead of ordering by how many hands *will* be of a given type (which depends on the ranking of the hands), just order by how many hands *could* be of the given type. Technically, this also applies to the original ranking. A four of a kind is also two pair and a three of a kind, if you think about it. The only reason we exclude them from the count of the lower hands is because they're lower hands, but the only reason they're lower hands is because they occur more often; it's the same circular reasoning, and it only pans out without jokers because of the structure of the deck. There happens to be a self-consistent ranking without jokers, but we just got lucky, and our luck didn't hold. Instead, update the ordering rule so that self-consistency is not necessary, and just abandon that each hand has to slot into only one category

There seems like there's a simple way to solve this, which is to rank hands as if you were using the joker to achieve that hand, and not the "best possible hand". That way, three of a kind is ranked higher, since 137,280 < 205,920.

Missed opportunity to call it the Joker Poker Paradox

Could you solve the paradox by ranking Joker + 1 Pair separately? There are 82,368 combinations of Joker + 1 Pair, so that would put it higher than Natural 2 Pair, and lower than Natural 3 of a Kind. The new ranking would be: - Natural 3 of a Kind: 54,912 - Joker + 1 Pair: 82,368 - Natural 2 Pair: 123,552

The paradox is that it's fallacious to argue that the rankings of hands, based on probability alone, should be affected by what you, as the player, would preferentially turn your joker into. Just because most players would choose to turn their pair and joker into a 3 of a kind instead of a pair and a random leftover, does not mean that the existence of that hand goes away. If i have two 2's, a 5, a 6, and a joker, it is indeed smarter to make it into a three-of-a-kind 2's, but i can still use it to make a 2, 2, 5, 6, 7, for example, if i wanted. So in reality, there isn't actually any paradox, and it is indeed possible to objectively sort the hands based on probability.

You can resolve this by counting combinations not as their highest rank, but by what they make. Aka a full house counts as a full house, 3 of a kind, 2 pair, 1 pair, and high card. This would resolve it, as changing what you choose to score it as does not alter what it counts as

Going in: "I can do anything!" Leaving: "Who keeps spinning the world around?"

It blows my mind how someone can make comprehensive math problem videos and hilarious sketches at the same time

The math is neat, but there absolutely is a way to order the hands based on probability. You order them based on the chance to get that hand, regardless of whether you would actually choose that hand.

The "Royal Flush" isn't a different type of hand, it's just the best type of "Straight Flush". You wouldn't call a "Pair of Aces" a different type of hand from a "Pair".

I have an old handheld video poker game where twos are wild, that I instantly thought of when seeing the subject of the video and tried to remember how it ranked them. I thought I remembered it making the payouts for them even, but upon checking it actually does that for four of a kind and full house, interestingly. I think they would have done the same with two pairs, but instead it's not even in the rankings! Five of a kind is actually under a royal flush, but keep in mind that this is with four wild cards instead of just one. The top three ranked hands you can get, in ascending order, are a royal flush using twos, a four of a kind of all twos, and then finally a natural royal flush. The payouts are also all very low, since you're winning a lot more often on average even without the two pairs even being listed. It's very interesting to see a three of a kind be the lowest possible payout, which just returns the same amount of points it cost to play a hand in the first place!

One rather interesting idea to fix this "issue" is to make the rankings of 3 of a kind and 2 pair the same. When comparing a three of a kind and a 2 pair, compare the rank of the triplet of the 3 of a kind with the higher pair in the two pair. The higher rank wins. In the case of a tie, the 3 of a kind automatically wins. Although I think the solution is just to keep the rankings the same, because as someone else commented. When counting probabilities, you should count a hand if there _exists_ a card that the joker can turn into that would create that hand. That way, it doesn't matter the ordering of the hands, the probabilities will stay the same.

My solution to this is that you include all hands that could qualify even ones that you would claim a higher category. I would do this even without wilds eg. A full house would also count as a three of a kind, two pair, a pair and high card, as it has all of those things.

The Joker is always wreaking havoc..

When you calculate the possible ways to get any state (such as two pairs) you use the former result without the joker. Thats not how it works since the former result is when used 5 cards but you are calculating 4 cars and then adding the possibles ways of the joker

it shouldn't be a paradox since the probability of getting a two pair and three of a kind shouldn't vary depending which you rank higher...sure, the probability a player plays said hand might vary but the number of ways to get each hand would still be 205920 and 137280 respectively and thus there is no paradox and nth should change

This Video (and "Riddle") is awesome, just one tiny tiny little thing: When you switch the Text "Three of a Kind" and "Two Pair", you do not switch the Cards to the exact right of it.

You will more often have the possibility of getting a single pair rather than a 3 of a kind, therefore ranking a 3 of a kind higher than a single pair. You may choose to turn a single pair into a 3 of a kind by utilizing the joker card, but that shouldn’t influence how the different hands rank against each other

Fun fact: If you're playing poker with a 32 cards deck, then a flush is less likely than a full house and even less likely than four of a kind !

Seems to me an interesting game would be a to deal out poker hands with wild card. Then have a bidding round (like in bridge) where players try to compete on who decides the ranking, Three of a Kind high, or Two Pair high. Maybe the stakes for bidding would be poker chips. Then we play poker based on the ranking chosen. Interesting because if you have one of those One Pair plus Wild card hands, you might care whether it becomes a Three of a Kind or a Two Pair based on what cards those are.

It seems like this should also happen with full house vs. 4 of a kind and one pair vs. high card, at least if you add in more than one wild card. There's another interesting phenomenon that if you deal out a card at a time until either a straight or a flush is possible from any five cards on the table then a flush is more likely to come up first even though it's a higher ranking poker hand.

The paradox only happens because the column is misleadingly named "possibilities". Those numbers involve more than just counting possibilities. There are 40 possibilities for getting a straight flush but only 36 of those would result in the 2nd-ranked hand. So it should be labeled "possibilities of having these cards while also not having a better, already defined rank". If you're saying you want to rewrite the definitions of rank based on probability, you have to stick to just using probability. What you're doing is recursively using the definition of rank while writing that very definition.

I still love how your videos are still interesting whether or not it contains aliens and/or Jigsaw

Seems to me like the solution is for two pair and three of a king to be equal in value and depend on their high card to declare a win or draw

Keep up the good work! much love from Ethiopia ❤❤

Very simple to solve this if you made a game of it. 1 - The non-wildcard hands are ranked higher than their wildcard ones. 2 - No 5 of a kind, as there are only 4 kinds of cards (hearts, diamonds, spades and clubs). 3 - Wildcard has to be discarded in the flop/turn/river, wildcard is only in play the hole cards 4 - Three of a Kind Wilcard / Two Pair Wildcard are treated as the same rank and only beat the One Pairs. 1 Royal Flush, 2 Royal Wildcard Flush, 3 Straight Flush, 4 Straight Wildcard Flush, 5 Four of a Kind, 6 Wildcard Four of a Kind, 7 Full House, 8 Wildcard Full House, 9 Flush, 10 Wildcard Flush, 11 Straight 12 Wildcard Straight 13 Three of a Kind 14 Two Pair 15 Three of a Kind Wildcard / Two Pair Wildcard 16 One Pair 17 One Pair Wildcard 18 High Card

I just came from class thinking I hate this shit, then I watch a zach star video and go yeah I f*ckn love math.

There is a solution, and that is to count the number of hand that could produce that result, even if it could also produce a higher result. This does make the probabilities sum to more than 1, since every hand has a high card, and full houses also count as both 3 of a kind and 2 of a kind, etc. but it does allow you to rank the hands consistently.

I believe this is only a paradox if the game rules dictate the joker HAS to be the card that makes the best hand. If the player chooses what the Joker represents, then you can adjust the right column to be calculated differently. Instead of iterating over all hands and adding one to the possibility count of the hand type with the highest ranking that hand can make, add one to ALL hand types that are satisfied by the given hand. For example, the hand (8H, 8S, 8C, 8D, 2H) would add one to the possibility count of High Card, Pair, 2 Pair, 3 of a Kind, and 4 of a Kind. Then rank them based on those possibility counts. Despite the total possibility count exceeding the total possible # of hands, I believe the hand types' corresponding possibility counts still correlate to rarity. This means 2 Pair and 3 of a Kind would have unconditionally defined possibility counts with which to be ranked. Is there a flaw in this ranking algorithm? Let me know if I missed something!

The paradox, if i'm understanding correctly, is that if we assume 3oaK to be the higher ranked hand, that will cause players to interpret all hands containing one pair+joker as 3oaK (rather than 2 pair), which flips the order of scarcity. The thing is though... so what? We should allow that scarcity order to be flipped, because that is not the order that matters. The only thing that matters is the scarcity in our *opportunity* to make a hand from a well shuffled deck, not the scarcity in outcomes post-interpretation. If 2 pair becomes scarce only *because* we choose to discard it, that tells us nothing about how difficult it is to construct that hand from a well shuffled deck, which is the measure that should be tied to ranking. It is always harder to construct 3oaK than 2 pair from a well shuffled deck, even with a joker, so that is the hand that should rank higher.

4:00 This would be better if you gave the number of "one pair plus joker" hands. Don't make me do subtraction on six digit numbers.

What I find paradoxical is the MDF from the bettor's perspective is always lower than the pot odds for the potential caller.

Three of a kind with a joker automatically becomes 4 of a kind too, and even more frequently than before full house.

the possible way to order the list is to just arbitrarily choose one and force a wildcard with one pair in hand to be used always to make 2 pair or 3 of a kind

Wow, amazing, loved the video and how the rankings go back and forth, truly something to be considered when adding wild elements to all kinds of games

A better visual representation would be using Venn diagrams. Have 2 circles that are intersected. One circle represents ways to get 2 pair, the other is ways to get 3 of a kind and the intersection is having the option of 2 pair or 3 of a kind with a joker. Assuming hands automatically become the higher ranked hand with the joker, the area in the circle that represents the higher ranked hand keeps the intersection and the other circle loses the intersection. If initially one circle is smaller than the other you rank it the highest, but that shrinks the other area, Hence the paradox.

With the invention of Balantro, a roguelike poker hand making game, the introduction of wild face cards, flushes with 4 cards, and so on, this suddenly becomes an applied problem. How much should you upgrade the two pair vs the three of a kind payouts if you obtain this wild card in your deck, including downtime where you do not have it and when you are allowed to mulligan your hand?

The paradox isn't caused by the joker, it is caused by the assumption that the probability of a hand should determine it's ranking. For example, in a regular deck there are 1,020 ways to get the worst hand in poker, a non-flush 7,5,4,3,2. We don't then say those hands must be better than the 3,744 full house hands.

Nitpick, but I hate seeing hand rankings with royal flush at the top. It's not a new rule, it's just a word for a type of hand. It would be like putting a "wheel" between three of a kind and straight or a "broadway straight" between straight and flush.

Some food for thought: What if the joker can only be card that is currently not in play? eg not in your hand, on the board or revealed by an opponent at showdown? This removes the 5 of a kind option. Also adds a new level of jeopardy where you may have to change your hand at showdown. Eg your hand is AAAK + joker. Your opponent reveals the final A in their hand so you get downgraded from four of a kind to full house

5:41 love that low quality 10♦

7:47 sierpinski triangle. Learned about that yesterday

Im seeing a lot of comments about how it would be pointless to calculate the results without regard for the "optimal" choice, as if there are never situations where you would choose to create a lower ranked hand. I can give a few that I've PERSONALLY done, and some I have seen others try: - Teaching little kids, where you might go easy on them the first few times so they don't get frustrated while learning. - You are playing on an app or in a casino where for a certain amount of time get a reward of some sort (free drink, game token, etc) for getting a certain number of each specific hand (ie, you have gotten 99/100 2 pairs necessary for a reward, but only 80/100 3 of a kinds). Assuming the amount you bet was less than what the reward is worth (say you bet 5 quid but the drink is worth 10), AND you won't be able to get enough 3 of a kinds before the reward period ends, you'd pick the 2 pair - You deliberately threw a match for some reason (other than teaching a kid). - You are trying to be clever by masking your tell. You have good cards, you flinched, the other players saw you. But you could deliberately tank the hand by choosing the lowest ranked possibility, so the opponents think that's your tell for BAD cards. Ultimately whether these things make sense or would work doesn't even really matter, because even if they aren't wise, someone with think they are and will try it. So TLDR, I think a video where you don't assume the player would always pick the better ranked hand would be worth doing

i think what should be done is compare them both as if they were both advantageous to the game result(ie. if the players were dummies) and compare the two numbers basically ignore the intelligence of the players

Would be nice to see a video about the probabilities and the information available. That is: if you do not know the hand of the other guys, you have 52 choose 5. However if you saw after the first card your mate got a 5 of clubs, this vastly changes the probabilities of your hand (you know you won't get that 5 of clubs). This is simple but would be very interesting to see explained with greater depth!@

this was way more interesting than i thought it would be (in a good way)

Meanwhile the joker laughs at us... makes you wonder who really is the jester...

There's also the added question of what if two players have the same hand, but one has the joker. Do they tie or should the joker have less weight? And should a natural 2 pair have greater or less value than a 3 of a kind with a joker?

What changes if we say a natural hand beats an equal wild card hand? So Ac Kc Qc Jc 10c beats Ad Kd Qd Jd Joker, but a 10-high straight with a joker still beats a 9-high natural straight. So higher hand still always wins, but a hand with no joker beats equally-ranked hands using a joker. This might remove some incentive to upgrade one way or another. Not ever sure how to analyze this into formulas.

What happens when you add more possible winning hands, separating out the wild cards? For example, one winning hand is a royal flush using the joker and a separate winning hand being the royal flush without using the joker.

While the paradox works on paper and in theory in practical use most of not all players will still rank the hands of poker in the same order if using a wild card because the likelihood of having the same hand is still relatively the same. 1 wild card doesn’t drastically change the outcome of each hand in practical settings.

I thought 5 of a kind was 'illegal'... When I was a kid, I thought we were taught that although the wild card could 'be' another card, it could only be a card that nonetheless existed SOMEWHERE in the deck. And since there are NOT 5 aces in a deck, you couldn't MAKE '5 of a kind'. The wild card could take on the identity of any EXISTING card...

You simply need to add the “one pair plus joker” combos to BOTH two pair and trips. It’s accurate and actually relevant in wild card variants of open face Chinese poker where you sometimes use a wild card to make a lower hand to fit within the 3-hand hierarchy.

Your conclusion doesn't lead to a paradox. When dealt a joker, it can be whatever you like. Thus, the sum total of the number of occurrences of two pair is still increased by the Joker, since the player is not compelled to use the Joker to create 3 of a Kind, even if that is most optimal. In practice, the player will choose the most optimal play, but that doesn't mena they are compelled to and thus it does not affect the mathematical probability based on the information or conditions offered

What about the high card? There is mayn’t ways to get 4 cards that don’t match and then get a joker, to make the hand a Pair. Would that be enough to make the likelihood of getting a single pair way higher?

Well, you correctly added "5-of-a-kind" at the top of the chart. The next thing to do is simply to remove "2-pair" and "high-card" from the chart, since 3-of-a-kind is better than 2-pair, we would never make a 2-pair hand (and a high-card hand automatically becomes a 1-pair hand).

I did the 0/1/2-Joker calculations about 20 years ago and realized this issue. That's why I ultimately never played with jokers.

There is another place where the paradox shows up. If you draw 3 of a kind + Joker + Other you can either make it 4 of a kind or a full house.

The ranking is defined by the rules of poker. There is no need to change the ranking. And therefore all possibilities can be calculated also for a set including one joker

Not a paradox, just bad logic. There are still more ways for your hand to be a two pair than a 3 of a kind. The fact that you would always want to turn your pair + joker hand into a three of a kind, doesn't mean you have to. You can still turn it into a two pair, also increasing that number of possibilities. If you want to calc the proper number of possibilities, you would need to count all transformations of the joker separately.

I'd like to see the numbers for simply replacing one of the cards with a Wild. For example, the 3h is now wild. There's only really three specific times it needs to be EXACTLY the 3h. A2345, 23456, 34567 of hearts. Other than that, it would either be the Ace (or next highest available) of hearts, or any other card in the deck. And any other time, the suit doesn't matter...at least in poker. Either way, very curious. Nice work!

you should change perception of hands into 4 cards versions so you can have 4 of a kind -> 5 of a kind 4 for a royal -> royal flush 4 for a straight flush -> straight flush 4 for a flush -> flush 4 for a straight -> straight a trips -> full house two pair -> full house pair -> trips Nothing -> pair so the only hands that are possible are on the right, and you should reorder these by probability (adding probabilities of full houses together)

In Pai Gow poker, a joker is used as a quasi wild card-can be a 5th ace, or used to make a straight or a flush. As a result, a straight pays less than three of a kind on a side bet for having a straight or better.

Ace to Five straight flush has the same odds as a royal flush.

This is not a paradox. The ranking is based on the possibility of that hand to show up, not which one the player would choose. When the joker gets introduced, both two pair and three of a kinds possibility to show up increase and two pair is still more likely, therefore its still a weaker hand.

Considering that the set of 1 pair plus joker always included in the higher group, then the natural 3 of a kind would still be higher. So the order should be the calculated order, except for the line of the 3 of a kind would be shown as natural 3 of a kind/1 pair plus joker.

Doesn't the same paradox exist between One Pair and High Card, considering how close their numbers were in the non-joker case?

Great video!

What if you just put "two pairs" and "three of a kind" in same rank so that it would be "two pairs or three of a kind"? Then if there were for example cards 3, 7, 10, 10 and joker then joker would be in superposition of 7 and 10 making it two pairs and three of a kind at the same time.

I'm just glad I noticed it instantly :D

You could take this idea to the extreme. Imagine a deck consisting of only jokers. Then if we have 5-of-a-kind as the highest ranked hand, then 100% of all hands would be [chosen to be] 5-of-a-kind, making it the most common hand and should therefore be ranked lowest. As other commenters have pointed out, the problem is the interdependency between the ranking by probability and the choice for the joker. We rank based on probability, but then the choice for the joker is based on rank, which then skews the probabilities (and the rankings).

With a Joker wildcard included there'd also be zero possibilities for a High Card hand because you'd always make a pair with your highest card of the four, with the Joker assuming the same rank.

You could resolve the paradox by combining the two kinds of hand into one category. (in other words, two pairs against three of a kind results in a draw, so only highest card value decides)

The possibilities don't change just because you wouldn't choose that possibility. The correct possibilities for the chart with a joker would be 137,280 (3 of a kind) and 205,920 (2 pair).

I don't see it as a paradox, I see it as a value choice.

I am not sure that logic at 5:14 fully checks out: counting the ways to get Two Pair could more reasonably involve only the deck AFTER the wild card has been changed into the best possible hand. That means Two Pair will not be chosen at all and is not part of the rankings anymore. It drops out of the set. A counterargument in the vein of "but, but why would people choose to make One Pair into Three of a Kind if they do not know what the real rankings are?" falls flat: you cannot have both simultaneously, at the same step of the process - a choice that determines the rankings and the rankings that determine the choice. That way you would enter a rather typical logical paradox like the classical set theory paradoxes. Still, such a paradox is not necessary. A paradoxical outcome sometimes simply means you got your processes not in order. In this case, it means you need to axiomatically establish a hierarchy of choices: a) you may either determine that the old rankings stay as they are and people choose according to them and ONLY AFTER such a choice you would determine what the new order is. Maybe still slightly paradoxical because such a new order might lead to different choices, but well, that is what our order of choice was for. That is the order that I have chosen in the first sentence above, rather arbitrarily, of course. I could have also chosen b): b) Or you may determine that you first see how many possible ways there are to reach a specific order, like Two Pair or Three of a Kind, and if more than one choice is possible, let all choices count for the total. Then make the rankings, creating a kind of singular pathway for people for those cases where before there was more than one option. All in all, the process might not be mathematically fully satisfying or completely elegant, but it gets the job done and avoids deliberately jumping into any unnecessary pitfalls. Those will only swallow you whole if you insist on the impossible.

The paradox is easily solved by using the number of possibilities that yield each result *regardless* of higher ranks achieved. Really, this should have been done from the start, since it was only by chance that "high card" happened to have more possibilities than "one pair". But it would be silly to rank "high card" higher than "one pair", given every single instance of the latter also includes the former.

Basic logic left the room here. The probability to create Two Pair or Three of a Kind with a Joker stays the same regardless of the ranking and how we want to use the Joker. This means Three of a Kind should always be with higher ranking because it is more difficult to get it with or without a Joker.

Very cool. Thanks.

What about when the Joker is not used, i.e. one of the other cards are made wildcards? E.g. the One-eyed Jack, Suicide King, deuces?

The order is very easy to determine. You look at the combos to do each option, see how many ways there are to do them, rank them. IF you were to try making Two Pairs it would give more options then if you tried to make Three Of A Kind.

Yeah, this can be solved with one extra dimension of analysis. Since jokers are not a random card, they are a free choice, all joker hands need to be counted in a separate tally from the purely random chance hands. You can model this by adding every possible joker hand choice (52 possibilities for each joker hand) to the counts. This will have the effect of dramatically inflating the counts, but will not change the order of the hands (99% sure of this base on thought experiment, not actual numerical calculations). This will exactly model what would happen if we treat the joker as just a random extra card, which is actually what we want when we are discussing the probability of each hand. The fact that we get to *choose* the best hand in actual gameplay just means a lot of those possibilities will be ignored due to the *not* random nature of free choice. That does not mean those possibilities are impossible, someone could choose to make two pair instead of three of a kind, but it is acknowledged that we are not in a pure random chance xCy scenario anymore. To put it differently; there are still more possible two pair hands than three of a kind, it's just that you would have to be dumb to *choose to make* a two pair hand instead of a three of a kind. That is an entirely different argument than there being less possible two pair combinations in total.

In the 3 of a kind v.s. 2 pair paradox, ranking a natural 3 of a kind v.s. one using a wild can be used.

Why does combinatorics break my brain lol. I guess I just need to practice.

it's amazing how many comments solved this paradox correctly. now i am happy.

My answer to the question posed at 1:47; you definitely need to add 5-of-a-kind to the rankings. I wondered about this question a few days ago, but for two decks.

the best way to resolve this is to play the best 6 card hand vs the best 5 card hand.

You need these tables for seven cards.

what if you keep natural 3 of the kind and two pair higher than hand with joker?

Since you started doing comedy I can't listen to this without eagerly awaiting some nonsequitur

As always, it is the case that paradoxes do not actually exist. They are only the effect of some sort of misunderstanding or incorrect assumption in our logic. In the case here you incorrectly assume that because one should not use the joker for a certain hand that one cannot do so. The possibilities must count everything one can do and in that case you find that three of a kind remains higher in the list then to pair regardless of the fact that one would not use the card for two pair. The fact that they can must be taken into account.

The simple solution is to do it off of all possible ways to get to the hand. I would argue that anyone that says this doesn’t work is simply doing so because they want this to be a paradox when in actuality there is a clear number of the actual probability that a hand could be formed, not what a person is going to choose as poker never has taken that into consideration but instead the actual chances that the hand can be formed.

finally one video of zach where I understood everything..

Easy solution: those two hands are now of equal value. You can differentiate them by if both pairs are higher than the three of a kind the two pair wins, and if the three of a kind is higher than both pairs it wins. If the three of a kind is between the two pairs, chop the pot.

I swear this vid got recommended to me because I watch a lot of people play Balatro

This is why in pai gow poker, jokers are only wild when completing a flush, straight or straight flush. In any other situation, it counts as an unsuited ace. Interestingly, it does introduce a new hand ranking, which is 5 aces.

Fixing the joker outcome to the optimal hand and calculating the odds creates an issue but allowing for the joker to be any of the 52 cards to calculate possibilities still gives a proper list.