Math With Dice – Part 1: Probability of Rolling Six Sixes

Dice are fun – you can roll one and quickly see a random output from 1 to 6 – without electricity! But it’s even more fun when you roll more than one. You can easily see that if we attempt to roll several dice and get one number – for example, 6 – on all of them, it becomes much, much harder every time we add a die. Assuming the dice are evenly weighted, the chance of this happening is cut in 6 every time we add a die. That means that with one die, the chance is 1 in 6. With two dice, it is 1 in 36. With three, 1 in 216. And with six dice, the chance of all of the dice rolling 6 is 1 in 46,656. That means if you rolled once every 2 seconds, it would take on average one day to get this combination. If, however, you only wanted to get 6 of the same number, the chance would be 6 in 46,656, or 1 in 7,776 – it would only take 4 hours. We can express this as (6^n)/6, where n is the number of dice, simplified to 6^(n-1). And just because, let’s say that we had sixteen dice, and we wanted to roll a six on all of them. That would mean that, on average, it would take 2,821,109,907,456, or 2 trillion, 821 billion, 109 million, 907 thousand, and 456 rolls of those 16 dice to get 6 on every die, because 6^16=2,821,109,907,456. If you rolled once every 2 seconds, it would take you 175,546 years on average to do so. To put that in perspective, humans have only practiced agriculture for around 10,000 years. It’s a long time.

Three red and three green dice, all of which rolled a six