Dice Game Probabilities

Every year, around September, the Filipino Chinese community plays a dice game. I didn't grow up with this tradition so it was a real shocker for me. I didn't understand how rolling dice could be so fun. Nikki told me, "What's not to like, you roll a dice and you get a prize!" Now, I'll admit, the whole affair really is fun. People can really get into the spirit of trying to affect the roll by touching someone who's been lucky, or rolling the dice slower or faster, or not looking while rolling. People's superstitions come alive when they roll the die with their left hand when they've had an unlucky streak rolling with their right.

Obviously, the best roll would get the first prize, second best combination gets the second prize and so on and so forth. The winning combinations are set in stone and one need only consult Google to get a copy of the winning combos. Common wisdom would dictate that the rarer combinations would get the better prizes (i.e., a four of a kind would have a better prize than two fours, for example). As such, when I asked a friend how many of each prize to prepare, she gave me a very easy way to remember:

• 1st prize - 1
• 2nd prize - 2
• 3rd prize - 4
• 4th prize - 8
• 5th prize - 16
• 6th prize - 32

Basically just double the previous number. She also said that this isn't set in stone and you can have more than 1 first prize or more of the 5th and 6th prizes as consolation prizes. I generally like to prepare 20 of the 5th prize and up to 50 of the 6th prize - you see, I'm quite unlucky in these games and so I want to at least stock up on the consolation prizes.

In the few years I've played this dice game, however, I've always noticed something odd. I noticed that the last prize to be snatched up is almost always the 4th prize (you need to roll a four of a kind to get this one). The third prize combination (three fours) somehow get rolled way more often than the fourth prize combination (four of a kind). I had a gut feeling there was something wrong. The math wasn't mathing. So I started doing calculations on the probabilities but I realized I had forgotten my lessons in combinations and permutations and so my gut feeling remained a gut feeling.

That changed when I started using ChatGPT as my personal assistant. I threw my question to ChatGPT and all I needed to do was check if the math was sound. It was indeed and I was in fact correct. In fact, winning the 3rd prize is 1.67x more likely than winning the 4th prize. I then asked ChatGPT why a higher premier was put on rolling 3 fours than rolling a four of a kind. The reply was pretty straight forward - symbolism trumps rarity. The number four was simply given more weight.

I'm not one to change up the order - this is tradition after all. So I thought the best way around this is to simple change up how many of each prize to prepare. Here is my proposal:

• 1st prize - 1 or 2
• 2nd prize - 3 or 4
• 3rd prize - 6 to 8
• 4th prize - 6 to 8
• 5th prize - 20 to 24
• 6th prize - 40 up

Most likely the 3rd prize will still be snatched up before the 4th prizes will be taken but at least the wait won't be that long. What do you think? Yay or Nay? Will my Chinese ancestors disown me for changing up the number of prizes? I hope not.