King Louis

[Nick709]

The game is King Louis, a form of five card draw in which kings are wild as well as the lowest card in each players hand is wild.
So my question is if you hold a duece is it best to keep it? For example you hold 2579K, should you draw 3 to the K2 because you have a better chance of hitting another 2? or draw 4 to your K hoping to net more wild cards that way? If anyone knows the math that would be great too.

[Biggle10]

AKA 'Kings and Little men' when played as a pot-matching game. No reason to keep a deuce since you're always going to have a 'smallest' card.

[Nick709]

Right, what I am asking is if your attempt is natuarally to increase the number of wilds in your hand is it more likely that you will pair a 2 by drawing three to a K2, or is it more likely that you will draw a low pair that is lower than your other to cards if you draw four to a K?

[Alex/Mugaaz]

Guesses: I think you should draw 3 to K2 instead of 4 to a King, but you should probably draw 4 to a King if your lowest card is above 7 (complete guess), but I've never heard of this game or done the math. Also, I suspect drawing to a straight or flush is virtually always wrong ,unless SF with 2 wild? You probably need Jacks full to bet for value after the draw vs someone trying to play well.

[Biggle10]

The other question is are you playing Five of a Kind as a valid hand.

[MarkGritter]

My guess is that you shouldn't keep the 2, but let's work it out. Rather than trying to figure out "how often will I get my lowest card to be a pair", let's calculate "how often will I make quads or better". (Drawing 3AAA is no worse than getting 337A.)

Starting with 2579K and drawing three to 2K:

This is the easy case. Let x = one of 3-Q or A (41 cards). Then the possible 3-card draws that give you quads or better are:

222: 1 combination
22K: 9 combinations
2KK: 9 combinations
KKK: 1 combination
PPP (P=3467TJQA): 32 combinations
PPP (P=579): 3 combinations
PPy (P=3467TJQA, y != 2PK): 1776 combinations (=8 * 4C2 * 37 remaining cards for 'y')
PPy (579): 684 combinations
22x: 123 combinations ( 3C2 * 41 )
2xK: 369 combinations
xKK: 123 combinations
2xx: 2460 combinations ( 3 * 41C2, note we didn't count 2PP earlier.)
xxK: 2460 combinations ( 3 * 41C2 )

Total = 8050 combinations
Possible = 16215 combinations (47C3)

Quads or better = 0.496

Starting with 2579K and drawing four to the K:

This is more complicated. For each possible lowest card (I'll use 3 here) we want to calculate the number of possibilities for:

3333: 1
333K: 12
33KK: 18
33xx (x=4-Q,A): 3996
3xxK: 7992
3xKK: 444
3KKK: 4
3PPy (y=x except P): 6768, note we've already counted 33PP and 3PPK
3PPP: 124

This gives

16683 combinations with 2 as the lowest card (again)
19359 combinations with 3 as the lowest card,
15299 combinations for 4,
8877 combinations for 5,
9477 combinations for 6,
5211 combinations for 7,
5041 combinations for 8,
2511 combinations for 9,
1911 combinations for T,
859 combinations for J,
207 cominbations for Q, and
35 combinations with A as the "lowest card" (AAAA, AAAK, AAKK, AKKK)

This gives a total of 85550 combinations in 178365 (47C4) possibilities, or 0.480 of getting quads or better.

So, it appears that holding onto the deuce really is slightly better. But any errors I made in the calculation could easily swing things the other way.

'2' is the most likely lowest number whether we draw 3 or 2 cards. Notice that in many cases where we would get quads by drawing four, we would also get quads by hanging onto the two. (Drawing one fewer only matters if it's the fourth card that pairs.) I didn't work out how often we get five of a kind in the two scenarios, though; there might be a difference.

[bigpooch]

[ QUOTE ]


PPy (579): 684 combinations



[/ QUOTE ]

You made a slip here: 3x3x38 = 342 combinations (you can
also tell this is too many from the above entry: 1776/8 =
222 per specific unseen rank, so 3x222 = 666, but for unseen
ranks, the number must be less than 666).

Also, you forgot to mention that a straight flush can be
made (it's better than quads). Suppose we start with
precisely Ks 9s 7h 5d 2c. There are then 31 spade sf's,
28 heart sf's, 32 diamond sf's and 46 club sf's for a total
of 137 straight flushes.

Thus, there are 7845 combinations that make quads or better
in the case of Ks 9s 7h 5d 2c and the probability of making
quads or better is about 0.483811286.


DRAWING FOUR
--------------

I think the best way to think about this part is to consider
regular draw and what the chances are of drawing four to
a high card and making one of: quads, full house, trips,
two-pairs or a pair. You can see that for each case in
regular draw, you will make at least quads in "King Louis".

It just so happens I have a table for drawing four for
regular draw:

C(47,4) = 178365 combinations

quads: 52
boat: 288
trips: 4102
two pairs: 8874
pair: 76392

The total here is 89708, so to make quads or better AND to
make at least a pair in regular draw, the odds would be
about 0.502946206.

Here, I haven't even computed the combinations that will
make a straight flush or better.

It's clear that you should draw FOUR to a king rather than
keep the deuce in "King Louis".

[Alex/Mugaaz]

If you redo the calculations to have only 2 kings left in the deck will that change it back to keeping the 2?

[bigpooch]

No.

From my calculations, if you start with Ks 9s 7h 5d 2c and
the deck is missing the Kh, you should still draw four to
the king.

I get the probability of making quads or better drawing
three to K2 as 6810/C(46,3) = 0.448616601.

Even forgetting about making a straight flush when drawing
four cards to the king, you will make quads or five of a
kind with a probability of 74528/C(46,4) = 0.456708644.