From Monopoly to Backgammon to Yahtzee, our first experiences with board games almost always feature sets of six-sided dice. They’re a great way to create some randomness and chaos in a game, but players have always struggled with the odds of rolling specific outcomes. How likely is a Yahtzee in three rolls? What are the chances you land on Boardwalk? Should you pay the fine or wait to roll doubles to get out of Jail for free?
One main dice mechanic involves rolling two dice and adding up the total. So you might start to recognize that the odds of rolling a 7 are higher than the odds of rolling an 8. Another mechanic involves rolling lots of dice and looking at individual dice. The odds of rolling a 5 on one of four dice are higher than the odds of rolling a 5 on one of two dice. Usually, a game picks one of these two mechanics. But some games like to play on the edge…
Let’s say you’re playing a game that gives you a single roll of two six-sided dice. In this game, you roll a specific result k if the sum of your two dice add up to k or if either of the dice is showing a k.
In this game, given a single roll of the two dice, what are the odds that you roll a 5?And is it more likely for you to roll a 5 or an 8?
➡ The Solution
Dice probabilities have haunted mathematicians since the 1600s, when gamblers like the Chevalier de Méré first began to roll dice for money. As soon as things move past one die rolled once, the math of probability can get a little wild.
Fortunately for all of us, problems with two dice can be visualized in just two dimensions. That means that instead of learning some complicated equations, we can create a simple graphic that shows every possible way two six-sided dice can be rolled. To make it easier, imagine that the two dice we’re looking at are different colors: blue and orange.
With two colors, we can see the Blue-4 and Orange-3 is a different outcome than Orange-4 and Blue-3, even though these would be impossible to tell apart if the dice were the same color!
Basic probability tells us that the odds of an event occurring are equal to the number of ways that event can happen divided by the number of possible events. So, we can easily count up the number of outcomes that add up to 8 and divide by the total numbers of outcomes.
The strange rules of this game means that we get a result of 5 when either the dice add up to 5 or if one of the dice is showing a 5. Which gives us a total of 15 outcomes!
Turns out, rolling a 5 in this game is three times as likely as rolling an 8. Games like Craps often change the rules to make it a little tougher for someone to memorize all the odds, which is why keeping this grid of 36 outcomes makes every game with two dice a little bit easier.