Patrick Honner’s Skew Dice post

Patrick Honner published this interesting post about skew dice today:

Statistics and Skew Dice

This part caught my attention:

I asked participants to propose tests for fairness, and then had them perform a test I had decided on ahead of time: roll the die 100 times and report the number of sixes. Before they began, I asked participants to consider how many sixes they would expect, and what numbers of observed sixes might suggest to them that the die was unfair.

The reason it caught my attention is that I went through a similar exercise when we insured Pepsi’s game show “Play for a Billion” back in 2003 and 2004.

The game involved 1,000 contestants selecting a 6-digit number from 000000 to 999999. Each contestant had one guess. Following their guesses, we selected a six digit number by rolling 6 10-sided dice. If any contestant had the number we rolled, a prize of $1,000,000,000 would have been awarded.

It goes without saying that we wanted to be sure that the dice were fair! If they were not fair, the contestants would have had a better shot of guessing the numbers that the dice would roll. Here’s a rough description (that did not make the final cut for the show) of my thought process in selecting the dice – I rolled them 15,000 times:


We were not the only ones to have concerns about the dice, though. Both Pepsi, the TV network, and the show’s auditors wanted to be sure that the dice were fair. So, all the dice that would potentially be used on the show had to pass an additional set of statistical tests designed by the auditors. It took about 5 hours of testing the day before the show to select the 50 dice that would be used on the show (out of about 2,000 dice that I brought). Here’s a short clip showing how we picked the number on the show.


So, a funny coincidence for sure, but figuring out of a set of dice is fair isn’t just a theoretical statistics exercise – in 2003 it was a real exercise I went through for $1,000,000,000 contest!



2 Comments so far. Leave a comment below.
  1. I’m curious whether you concluded that 1950 of the 2000 dice were actually unfair? Was the distribution of guesses by contestants roughly uniform or were there any interesting patterns?

    Also, would it have mattered if the dice were biased, even known to be biased, but if the nature of the bias was not known?

    • It wasn’t that the other dice were deemed to be unfair, it was just that they didn’t pass the statistical test designed by the auditors. The passing vs. non-passing dice could have passed the test just by chance, though.

      Their concern was that I could have somehow rigged the dice to avoid throwing numbers that people were likely to guess. Since I knew that I wasn’t cheating (not to mention the difficulty of avoiding numbers that you don’t actually know!) I wasn’t particularly concerned about their test. I honestly don’t even remember what it was.

      The dice we used were certainly not casino grade dice, but I was comfortable using them for a single roll. The bias in the first set of dice we tested was obvious enough that you could have increased your odds of guessing pretty substantially if you knew we were using them. The second set was random enough that I was confident no amount of study would improve your chances by any meaningful amount on one roll.

      As for the contestant numbers, I played around with them a little and wish I would have saved that work. My recollection was that the guesses had a lot of what you would expect – dates, for example – and the individual digits were definitely not random. The first year of the show the winning number was 1 7 8 2 3 8 and I was confident that no one won. The second year, though, the number was a date 1 1 2 7 3 4, I think, and I was a little scared that that was going to be someone’s birthday.

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