Playing 5 card draw with a stripped deck of just faces, aces and a joker (17 in all) Are my odds ok?

The deck has 17 cards. The joker can be Ace, King, Queen or Jack. My percentages for each hand are the following:

6188 permutations

0.06% - 4 - Five of a Kind
0.06% - 4 - Royal Flush
3.88% - 240 - Four of a Kind
4.07% - 252 - Straight
8.14% - 504 - Full House
24.82% - 1536 - One Pair
27.93% - 1728 - Two Pair (also, rarer than a one pair?)
31.03% - 1920 - Three of a Kind (most common!)

Flushes have to include the joker, which I guess then represents the 10. But it's more of the fact you have one of every available card. These odds also only relate to the initial draw, so maybe the one pair/two pair/three of a kind mismatch would even out after replacing cards. Not sure. Can anyone help with the odds for the initial draw, and then for the final post-replacement hand?

I think the game ends up being surprisingly deep.

Your comment that two pair is rarer than one pair is backwards and the numbers reflect that. ONE pair is rarer.

Why is one pair rarer?

Think about it. Let's start with 4 cards, containing one pair. Say, A-A-K-Q.

That fifth card has to be a Jack to be one pair. If it either a King or Queen, it's two pair.

4 outs vs 6 outs. With a full deck, there are 40 out vs 6.

Interesting poker variant.

Assuming this is 5 card poker and assuming that straight > flush, I do get the same number of combinations and probabilities for hand categories as you have posted.

1. How many cards can a person draw?
2. Is the draw decision to be optimized against a payout table (or scoring table) or is a player trying to beat a single opponent who can also draw?
3. Are the poker hand rankings intended to be as shown on your table above, by frequency of hand? It is a paradigm shift to rank a three of a kind and a two pair lower than a one pair hand. Usually people draw to make their hand more coordinated - more highly paired or more highly suited or more highly "connected."

And, I'm confused. How is "odds for the initial draw" different than "odds for the final post-replacement hand"?

One problem with your poker hand rankings is this:

Just as a Royal flush will always be interpreted as the highest possible ranked hand; i.e., a royal flush as opposed to a straight or a flush, - a 3oaK could be interpreted as a One Pair hand where one of the kickers has a rank that is the same as the pair.

Sorry I'm not being clear.

1. You do one round of drawing, once, for any or all of the five cards. While playing I've found there's no reason to ever exchange more than three.
2. It's just 1v1, player vs player to get the better hand.
3. The hand rankings refer to frequency alone, which is why it's weird that to get a "better" hand you're actually going to go against the odds. My question kind of asks that if a player intends to get a better hand (such as three of a kind over two pair), would the odds of drawing cards that give you that hand be higher than that of a lower hand?

By the odds initial/vs post replacement, I mean

My odds given in the first post = the first 5 cards odds
The odds of a hand after taking into account drawing replacement cards = post replacement.

Edit: double post

Okay, but let's consider this 5 card hand:

AAQJ-Joker; you and I calculated this hand to be a 3oaK, but shouldn't the joker be used as a King to make this a higher ranking One Pair hand? And, if so, doesn't the One Pair Hand become more common than the 3oaK hand?

You may want to define Trips>2Pr>1Pr no matter how often they might occur.

EDIT: Never mind, I just read your post above. So with the exception of Straight>Flush, your hand rankings are the same as traditional poker hand rankings?

1. With only 17 cards, you really can't deal 5 cards to two players, and then allow each of them to draw up to 5 cards-unless you require the 2nd player to draw from the mucked cards of the 1st player.

2. In a Player vs Player game, you need to realize that the presence of the cards in your hand changes the possibilities of what your opponent has.

Particularly in a 17 card deck with a joker, if your initial 5 card hand has no joker then you are playing against a stronger distribution of hands then you calculated for a fresh 17 card deck, and vice versa.

For example, you are dealt QQQ-JJ, a full house. Let's look at the scenario where you discard the two jacks and draw to QQQ and happen to get the other two jacks, so your post-draw hand is still QQQ-JJ. I believe that the probability of winning with your final hand, QQQ-JJ, is zero. If the opponent's 5 cards do not contain the 4th queen, he stands and beats you. If the opponent's initial five cards do include the case Q, he discards it, draws a card and beats you.

3. In fact, just as in 5-card draw, the optimum decision may depend upon upon whether you are drawing first, or drawing second and the number of cards your opponent has elected to draw.

Quote: Xenochria

The deck has 17 cards. The joker can be Ace, King, Queen or Jack. My percentages for each hand are the following:

0.06% - 4 - Royal Flush

As stated, a Royal Flush is impossible. I assume the Joker can be a 10, in which case, does an unsuited A-K-Q-J-Joker count as a straight?

Quote: DJTeddyBear

Your comment that two pair is rarer than one pair is backwards and the numbers reflect that. ONE pair is rarer.

Why is one pair rarer?

Think about it. Let's start with 4 cards, containing one pair. Say, A-A-K-Q.

That fifth card has to be a Jack to be one pair. If it either a King or Queen, it's two pair.

4 outs vs 6 outs. With a full deck, there are 40 out vs 6.

Yep, I agree, one pair is rarer, which obviously goes against standard poker.

Quote: gordonm888

One problem with your poker hand rankings is this:

Just as a Royal flush will always be interpreted as the highest possible ranked hand; i.e., a royal flush as opposed to a straight or a flush, - a 3oaK could be interpreted as a One Pair hand where one of the kickers has a rank that is the same as the pair.

My calculated odds take this into account, if it's a 3 of a kind, it doesn't count as a one pair. It's the best hand out of the combination of 5 cards.

Quote: gordonm888

EDIT: Never mind, I just read your post above. So with the exception of Straight>Flush, your hand rankings are the same as traditional poker hand rankings?

That's right, all straights are also a flush purely because there are only 5 different cards in the deck. (4 aces, 4 kings, 4 queens, 4 jacks, one joker). All the rest are the same as standard poker, but with different rarities.

Quote: gordonm888

1. With only 17 cards, you really can't deal 5 cards to two players, and then allow each of them to draw up to 5 cards-unless you require the 2nd player to draw from the mucked cards of the 1st player.

2. In a Player vs Player game, you need to realize that the presence of the cards in your hand changes the possibilities of what your opponent has.

Particularly in a 17 card deck with a joker, if your initial 5 card hand has no joker then you are playing against a stronger distribution of hands then you calculated for a fresh 17 card deck, and vice versa.

For example, you are dealt QQQ-JJ, a full house. Let's look at the scenario where you discard the two jacks and draw to QQQ and happen to get the other two jacks, so your post-draw hand is still QQQ-JJ. I believe that the probability of winning with your final hand, QQQ-JJ, is zero. If the opponent's 5 cards do not contain the 4th queen, he stands and beats you. If the opponent's initial five cards do include the case Q, he discards it, draws a card and beats you.

3. In fact, just as in 5-card draw, the optimum decision may depend upon upon whether you are drawing first, or drawing second and the number of cards your opponent has elected to draw.

1. As I said before, there's no reason to ever draw more than 3 cards so this never ends up happening. Even if it did, you wouldn't be able to trade with mucked cards.

2. Exactly, this is where there's a bit of strategy. For example, if you have the joker, you KNOW that the other player doesn't have it, whereas they might think they can get it from the draw pile.

3. Re: betting, the rule is: There will be two rounds of betting and after the first bet you can exchange cards in your hand for new ones from the deck; the highest bet gets first exchange.

Quote: ThatDonGuy

Quote: Xenochria

The deck has 17 cards. The joker can be Ace, King, Queen or Jack. My percentages for each hand are the following:

0.06% - 4 - Royal Flush

As stated, a Royal Flush is impossible. I assume the Joker can be a 10, in which case, does an unsuited A-K-Q-J-Joker count as a straight?

A Royal Flush is defined as an AKQJ of the same suit, and a Joker. Unsuited, it is a straight, which is also by the nature of the limited cards, a straight flush.

Quote: Xenochria

1. As I said before, there's no reason to ever draw more than 3 cards so this never ends up happening. Even if it did, you wouldn't be able to trade with mucked cards.

Why not simply stipulate that players are limited to drawing 3 cards? If you allow them to draw 5 cards, then someone will occasionally do that -whether it maximizes their Expected Value or not. Then, if the 2nd player wants to draw 5 cards he can't -there are only 3 cards left. So, what happens? Your rules should cover all scenarios, not just scenarios that are mathematically optimal,

Also, its impossible to analyze or simulate a game with a computer if the rules allow 4-card and 5-card draws but there are not enough cards in the deck to allow both players to make 4 and 5 card draws.

By the way, after a quick look, I don't see any strategic ambiguities with this game.

- Draw 0 to a 5oak, Royal Flush and Straight
- Draw 1 to a 4oak
- Draw 2 to trips
- Draw 1 to Two Pairs
- Draw 3 to One Pair

Never draw to a royal flush draw or a straight draw, always draw to your paired cards (including the joker.)

I think these strategy rules pretty much apply to every possible player hand - wild card or no wild card, 4 suited cards or rainbow, and whether you draw 1st or your opponent draws 1st and discards 0,1,2, or 3 cards. Also, whether your pair is JJ or AA.

Am I missing something? Is there a specific hand where you think there is a close-call decision as to how to discard and draw?

Edit: I do see one intriguing hand with a difficult decision: when you have AAJJ-Q, and opponent goes first and draws 2. You know opponent started with trips. Do you discard the Q and draw one card to AAJJ or discard the JJQ and draw three cards to AA? Its not obvious -especially because in many scenarios your opponent has either improved the trips or otherwise used up many of the cards that you need to improve.

Quote: gordonm888

Why not simply stipulate that players are limited to drawing 3 cards? If you allow them to draw 5 cards, then someone will occasionally do that -whether it maximizes their Expected Value or not. Then, if the 2nd player wants to draw 5 cards he can't -there are only 3 cards left. So, what happens? Your rules should cover all scenarios, not just scenarios that are mathematically optimal,

Also, its impossible to analyze or simulate a game with a computer if the rules allow 4-card and 5-card draws but there are not enough cards in the deck to allow both players to make 4 and 5 card draws.

I agree, it should be 3 cards to exchange max.

Is there a way to simulate this game with a computer at all?

This game reminds me when Ultimate Bet offered "Royal Holdem". The thoughtful players crushed the game early and quickly. Kings Full was roughly the median hand in that game.