# Badugi Poker Probability Odds

Knowing the odds and the probabilities of successfully drawing the right cards is very important in badugi strategy. In fact, there are times when it might just be more convenient and mathematically correct to just fold on certain hands instead of chasing after one with slim odds. First we will go through a few general concepts about probabilities and then I will add a table which shows a chart of different percentages of chance for different scenarios. One general concept you should understand is that when your hand is bad, the odds of improving it are better. This is sort of obvious because if you have 3 bad cards, you can choose to draw three more, so there is a good chance of getting something good.

## Odds of improving 3 card hand into 4 card badugi

This is one of the most common situations you will get into. Many times, players will get a 3 card hand and will be drawing for that last card to make it into a 4 card hand, known as a badugi. Let's assume that you have two suited cards and you need to discard one of them and try and draw for the missing suit that you do not hold. Well, there are 13 cards of a single suit in a 52 card deck and you already know what 4 of them are, so you are now choosing from 48 unknown cards. It turns out that the odds of being successful and choosing one card of a particular suit is 27%. This is roughly a 1 in 4 shot of getting a badugi. Be careful and read on because this is not the whole story and your odds get even less.

Remember that if there are a lot of players, such as 8 at a table, a three card hand is rarely going to win so you are always going to have to draw for a 4 card hand instead. Hopefully you have gotten one right off the starting draw though. Now in the above example, we neglected to factor in the possibility of drawing pairs. So not only must you draw 1 card out of 13 of a particular suit, you must not pair 3 of them. So you actually only have 10 possible cards that can improve your hand. These are known as **outs**.

Let's use the hand shown in the image above: A-2-3-4. The 2 and 3 is suited, so you would discard your 3 and draw for the missing suit, which is hearts in this case. At the same time, you must not draw an Ace, 2 or 4 of hearts if you are to successfully make your poker hand. So you start off with ten outs and the probability of drawing one of these "out" cards depends on how many rounds of draws are available. The maximum number of draws that a player can take in a round is three.

## Badugi odds calculator and equation for calculating percentages

So to find the odds of drawing the right card, we use a special equation to calculate the percentages. If you have a calculator, you can use the formula or equation above. There are two variables you need to know: how many out cards there are and how many draws you have left. So multiply 2 by the number of out cards and multiply that again by the number of draws you have remaining in order to find out your percentage and chances of making a 4 card badugi!

I know most folks are not into equations and do not want to get into algebra, so I made a table of every possible combination or situation that you will run into. So for example, if there are 8 cards still out and you have 2 draws remaining, then you have a 32% chance of making a 4 card hand out of a 3 card hand. The whole chart is shown below. You might be asking how do you know how many cards are out? Well, the only way you have any control over finding this information is if you have been dealt a card of this suit and you discarded it. For instance, in the hand shown above, let's say you drew a King of Hearts and discarded it in hopes of getting another lower hearts. Then you know for certain that there are 9 outs. In another example, if you discarded 3 cards suited hearts, then you only have 7 outs left. Even though someone else likely has a few of these suits, you just have to assume 10 outs if you have not been dealt a hearts suited card.

Outs | One Draw | Two Draws | Three Draws |
---|---|---|---|

1 | 2% | 4% | 6% |

2 | 4% | 8% | 12% |

3 | 6% | 12% | 18% |

4 | 8% | 16% | 24% |

5 | 10% | 20% | 30% |

6 | 12% | 24% | 36% |

7 | 14% | 28% | 42% |

8 | 16% | 32% | 48% |

9 | 18% | 36% | 54% |

10 | 20% | 40% | 60% |

Some of these figures are not realistic since you will never be in a situation with just out and 3 draws remaining, so some of these are just theoretical curiosities. Usually, a player will have to deal with 9 and 10 out. If you look at the chart, the odds are quite good if you have three draws remaining in order to turn a 3 card hand into a 4 card badugi. Just like I said at the beginning of this article, it is obvious, but now you actually know the numbers and the actual probability. Note that the equation and chart provides *approximate* percentages, so these are good estimates and not exact but they are highly accurate for practical use.

## Chances and odds of improving a 4 card badugi hand

Trying to improve a 4 card hand is dangerous, risky and sometimes outright greedy, but that's the name of poker! The only time someone would want to do something like this is when one of the cards is very poor like a king or queen. Calculating the odds is actually very simple. Basically you use the same chart shown above to find the probabilities using one less card of a particular suit. So if you were dealt a badugi on the starting flop and you discarded one card of a particular suit, then there are 9 outs. Then you just factor this into the table above. In fact, the best odds you can get when trying to improve a 4 card hand is right after the starting deal when you have 3 draws left. In fact, your odds are 54% of being successful of drawing a better hand. This is not as bad as it seems and can be a useful advanced strategy to try out.

If you are going to risk trying to improve an already decent hand, you should be watching the other players and trying to get a read on how strong their hand is. If they are drawing a lot of cards, then it might be smart to stand pat. If you think someone has a badugi, then maybe it would be wise to take a chance, especially if your high card is a king, which makes up the worst 4 card badugi hand. Remember though, any 4 card hand will beat out any 3 card hand, so this is a risk to take. I would recommend doing this only before the first draw when your odds are still somewhat decent. If you do not have any face cards, then it is already a very decent hand so I probably wouldn't risk it in this case.