Analysis - Ura Rates by Unique Tiles in Hand

I've seen enough people asking about the chance of getting ura dora that I decided to check the Houou replays to see the chance.

We'll first look at the four-player numbers. If we think about it purely mathematically, the ura dora tile is 1 of 34 different tiles. So, if the average hand has 11 tiles, that should be 1 - ((33/34) ^ 11), which equals a 28% chance of getting at least one ura dora, or a 72% chance of not getting any ura dora. In reality, the chance is probably higher, because lots of honors will die off in the early rounds and honors won't often be present in riichi hands in large numbers.

Let's check the data. First, let's just see the chance of getting no ura dora from your riichi, based on the number of indicators and unique tiles.

These numbers make sense. Using the total for 1 indicator as a baseline, if the chance of not hitting ura is 69.5%, you'd expect the chance of not hitting ura with 2 indicators to be 69.5% * 69.5% = 48.3%, so it's pretty close. The expected value for 3 indicators is 33.6%, and for 4 indicators, 23.3%, so these are all mathematically reasonable.

As the number of unique tiles goes down, the chance of missing ura goes up. However, the chance of hitting multiple ura will also go up. Let's look at the average ura to see the effect of this.

The data's rather thin for three and four indicators, but we can see that having fewer unique tiles, on average, results in about the same amount of ura dora as a hand with lots of unique tiles. How interesting. With 7 unique tiles in particular, that will usually be a chiitoitsu, so hitting 2 ura is a highly likely outcome for 7 unique hands. You can see this in the full data, which I'll dump at the end.

You may also be wondering why it stops at 5 unique tiles. A hand with four identical sequences would have 4 uniques. To be honest, I forgot. There are 0 four-unique hands in the sanma data though, so it's probably not important. Speaking of which, let's move onto the sanma data, starting with the chance to not hit ura.

If we do the same estimation as before, it would be 1 - ((26/27) ^ 11) = 33.9% to get one or more ura, and so, 66.1% to miss. That continues to be 43.6% for 2 indicators, 28.8% for three, and 19.1% for four.

The numbers for one to three indicators are around what was expected, but 4 is super low. This is because there are only 37 cases of having four indicators in sanma, three of which missed dora. With such a low sample, being widely off isn't surprising.

Let's also check out the average ura.

Again, the data for 4 indicators is very thin.

Finally, I'll dump all the data here for your viewing pleasure. First, the hanchan data. Not sure why the size is weird, blogger changed before this post and now it doesn't have an "original size" button.

Next, the sanma data.

So, now you know.