Mid-Autumn Festival Dice Game Probabilities

Ah, the familiar sight of dice rolling on a ceramic bowl.

It's the time of the year when Chinese restaurants or function rooms are filled with groups of people throwing dice into a bowl. For those who are unfamiliar with this tradition, the Chinese-Filipino community (among other overseas Chinese communities) celebrates the Mid-Autumn Festival by hosting social gatherings to play a traditional dice game. Basically, you take turns rolling the dice, and you can win prizes if your rolls match certain winning combinations.

I played my first game of the year last weekend, and I noticed that the 4th place prizes were the last ones to be consumed. This seemed to happen often in previous years, so I wanted to know if it was just coincidental or if there really was something "off" in the prize distribution probabilities.

For those who are not familiar with the game, here are the winning combinations and the corresponding number of items/prizes available per prize:


1ST PRIZE (in descending order): 1 pc

2ND PRIZE: 2 pcs
 

 

 3RD PRIZE: 4 pcs

 

4TH PRIZE: 8 pcs

5TH PRIZE: 16 pcs

 

6TH PRIZE: 32 pcs

 

To satisfy my curiosity over the events of last weekend, I computed for the theoretical probabilities of getting a particular prize in a given dice roll. For the math/stat-savvy readers, you may want to take a look at the computation logic in the appendix. I think the numbers are right, since I also ran a simulation in Excel that consistently gave roughly the same numbers. If you want, you can get the Excel file by following the link in the Appendix.

Here are my results:

One thing stands out:
The chance of getting the 4th prize is lower than the chance of getting the 3rd prize!

Okay, but does this justify why the 4th prize is usually the last to be depleted? Not quite. But from this, we can do the following (don't worry, this math isn't hard!):

As an example, if you have something that occurred 50% of the time, you can say that it occurs every other time (every 2 rolls). If you had 4 prize items to use up, and you use up 1 item every 2 rolls, it will take you about 8 rolls on average to deplete your items.

Running these equations on the probabilities presented earlier, we have:

Now, the 4th prize really stands out. On average, the 4th prize takes more than twice as long as the other prizes to be depleted. So don't roll the 4th prize too early and try to get other prizes first! There will be lots of time for you to try and get the 4th prize later on.

Other interesting take-aways from my analysis:

• You should expect to win something for 7 out of every 10 rolls (not considering the availability of the prizes). Now you can gauge how lucky or unlucky you are at your next dice game!
• Simulating stuff in Excel is fun!

Happy rolling! :)

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If you want an interactive summary of everything I discussed, check out this simple dashboard! 
  

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Appendix:

Excel File
https://www.dropbox.com/s/1pbys6u7pg4z0p7/Mid-Autumn%20Dice%20Game%20Probabilities.xlsx?dl=0


Prize Probability Calculations
Note: This section assumes that you have some background in counting principles and probability. 

1ST PRIZE

2ND PRIZE

3RD PRIZE

4TH PRIZE

5TH PRIZE

6TH PRIZE

Credits:
- Joseph Yap for the dice photo header.
- Fu character image used in the dashboard was taken from here:
http://www.ablogtowatch.com/panerai-says-fu-with-limited-edition-pam336-watch-for-china/