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Badugi: Odds of Improving a Four Card Hand

6 July 2009, By: Pokerjunkie.com
You have a four-card Badugi hand. Should you draw a card to try and improve your hand?
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This is a case of comparing risk and reward. When is the risk of making a worse hand bigger than the potential profits of improving?

If you draw a card to a four-hand, the risk consists of either drawing a higher card for a worse four-card hand, or getting an even worse three-card hand by either pairing the board or making a two-card flush.

How many cards are good for you?

As a matter of act, the only cards that can improve your hand are cards that are lower than your highest card without pairing or flushing the board. So it needs to be a card of the same suit as the card you discard and of different rank than the three you keep.

This doesn't leave us many outs. To be completely correct, the number of good cards is x-4, where x is the rank of the high card that you're about to discard. For example, if your high card is a Jack, x = 11 and there are 7 cards that will make you a better hand.

Odds of making a better hand

Since we've seen four cards, 48 unseen cards remain. Of these, the ones that are not good are by definition bad. This gives us the number of bad cards as (48 - number of good cards) = 52 - x.

In the best case scenario, your highest card is a king and so x = 13. The number of good cards is 9 and the number of bad cards is 39. The odds against you improving are 39 to 9, that is, around 4 to 1, or 20%. If you have more draws, the chances increase.

The table shows the chances of improving for all possible high cards in a four-hand (x), for one to three draws:

Discard

Good Cards

Chance %
1 draw

Chance %
2 draws

Chance %
3 draws

K

9

19

34

46

Q

8

17

31

42

J

7

15

27

38

T

6

13

23

33

9

5

10

20

28

8

4

8

16

23

7

3

7

12

18

6

2

4

8

12

5

1

2

4

6

(If your highest card is a four, there's no way you can improve.)

As you see, you're always a big underdog to improve your four-card hand in a single draw. In the best case only 19%. If you can make more draws, the chances go up, but of course so may the costs of staying in the hand.

Even if you get to draw three times, you're an underdog to improve even if you have a K high four-hand. But the pot odds and implied odds may still motivate a draw, of course.

(here we've disregarded the cards in the hands of opponents. If several opponents seem strong, your chances ought to go down a bit, theoretically. Since they are then likely to hold some of the cards you're drawing for)

Comparing odds

You must also compare the winning odds for your present hand to the winning odds if you improve. Improving to a 7 high may not win you the pot, for example.

All the while you need to keep an eye on the size of the pot - the pot odds. If there's a lot to be won, risks become more worth while. If the pot is small, drawing to improve a four-hand may not generate a big enough reward.

Your decision from here will be based on the actions of your opponents. If an opponent seems very strong, for example by standing pat and/or radiating confidence, your chances of winning with your present hand might seem slim, in which case drawing to a better four-hand may be your best plan.

That is, if you haven't already folded, which might be the best choice in such a situation. Anyway, you're probably going to fold to the next bet unless you improve to a pretty good hand.

On the other hand, if the opponent draws a card, you know from the above that he's most likely to have a three-hand after the draw. In this case, even a bad four-hand is now likely to be the best hand, and it would certainly be a bad idea to ruin it just for the tiny chances of improving.

Pot odds and implied odds

The more you can win IF you make a really good hand, the more likely you should draw a card. If you're on the first draw and you have three good low cards and/or several players remain in the pot, drawing and making a really strong four-hand may win you a big pot, while at the same time a lesser four-hand which is currently in the lead can very well be caught up and beaten by an opponent drawing to improve his three-hand.

On the contrary, if you're on the third draw against one opponent, the "leverage" is generally much smaller (the number of further bets you may win). Drawing may now not be worthwhile, depending on the size of the pot and the expected size of coming bets.

On the otherhand again, when you're at the last draw the implied odds disappear, so it's weasier to judge the situation and the risk-reward ratio.

As always in poker, everything comes down to "feeling", a sense of where your best chances lie. But the table above should give you a hint of the odds of improving a four-hand in Badugi, and your feeling can take it from there.

/Charlie River

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