Today is Dice Day. Actually it is National Dice Day, a US-only event, but what the heck, I wasn’t able to find a more interesting international celebration for today, so we’ll just drop the ‘national’ and internationalise it for now, if you don’t mind.
A die (or dice, in the plural) is essentially a small object with multiple sides/resting positions, and with different numbers or symbols inscribed on its various sides. It is designed such that, when it comes to rest after being thrown, each resting position has a main, opposite upper side. The number/symbol on this side represents the number of the throw. Dice are often used in gambling games and in many popular board games.
The most common die is a standard, 6-sided cube, usually with slightly rounded corners, with a number of dots (ranging from 1 to 6) inscribed on each side. When thrown, each side of the die is equally likely to end up on top, the result thus being a random integer between 1 and 6. For specific games, multiple dice may be thrown at the same time, with the outcome given by the sum or combination of numbers on the dice.
Of course there’s no reason that a die has to be a cube. Various other equally valid shapes exist – tetrahedrons, octohedrons, dodecahedrons, and more. When you think about it, even a coin used in a coin-flip is essentially just a two-sided die.
Dice are not only useful for gambling and board games – they are also a very effective tool to teach and explain some rather complicated statistics, like conditional probability, that is, the probability that an event will occur given that another event has occurred.
Take the example where we throw two standard, 6-sided dice. The probability that the second die will land on a 1 is 1/6 (it can land on one of six values, all equally probable – simple enough). Now, what would be the probability that the second die lands on a 1, if we know that the first die landed on a 1? The answer is still 1/6, because the two events are independent.
Things change, however, if the events are no longer independent. For example, what is the chance that the total of the two dice will be less than 4, given that the first dice landed on a 1? Firstly, there are six possible values (1,2,3,4,5,6) for the second die given that the first die landed on a 1. Of these, however, only two options, 1+1 and 1+2, result in totals smaller than 4. So, the chance of the total of the two dice being less than 4, given that the first die landed on a 1, is 2//6, or 1/3.
Formally expressed, p(total < 4 | die1 = 1) = 1/3.
There you have it – just when you thought you were safe in the knowledge that we were going to stick to chatting about harmless things like gambling and board games, you got flashed by a lesson in statistics…
Sneaky little buggers, these dice! 🙂