2 Apr

The Fascinating Science Behind Yahtzee

How likely are you to roll a Yahtzee, or any point-worthy combination, in one to three rolls of the dice?

Science of Yahtzee Math

At a glance, Yahtzee may appear to be one of the simplest games of chance you could possibly partake in. You have five dice, and up to three rolls to achieve a number combination worthy of scoring. Shake ’em, toss ’em and hope for the best, right? Chance – the noun, not the 5-dice total – plays its part in the game, of course, but there’s so much more to it than that.

The game of Yahtzee is deeply rooted in scientific behavior and mathematical probabilities. It’s fascinating! Not because knowing the odds of rolling a Yahtzee will help you make better decisions (although it can), but because the odds of rolling Yahtzee change every time you touch the dice; the odds of rolling any score-worthy combination of dice, for that matter.

The Science of Yahtzee

If you had just one die in play, the odds would be simple to deduce. There are 6 numbers, therefore a 1-in-6 (1/6) chance of any number being rolled. Increase the number of dice to 2, and you multiply the possibilities per die with each other. 6 chances on one die. 6 chances on a second die. 6 x 6 = 36; a total of 36 possible outcomes when rolling two dice.

Expand the formula for…

3 Dice: 6 x 6 x 6 = 216 possibilities

4 Dice: 6 x 6 x 6 x 6 = 1,296 possibilities

5 Dice: 6 x 6 x 6 x 6 x 6 = 7,776 possibilities

With this simple formula, we find that there are 7,776 possible outcomes on a single roll of the dice. Knowing this, we can learn so much more about the game and its scientific probabilities.

Odds of Rolling Yahtzee

Everyone wants to roll that perfect Yahtzee! It’s worth 50 points, after all – 100 more if you repeat it in the same game! So what are the chances of rolling a Yahtzee on 5 dice? We find this by starting with our original data, which tells us there are 7,776 possible outcomes.

Now, how many outcomes will give us a Yahtzee? Six of them:

1-1-1-1-1, 2-2-2-2-2, 3-3-3-3-3, 4-4-4-4-4, 5-5-5-5-5, or 6-6-6-6-6

So, the equation is 6/7,776 = 7.716, which rounds to about 0.08%.

But wait! This only gives us the odds of rolling Yahtzee in one roll of the dice. That’s no easy feat! What if we roll 2 or 3 of the same number in the first roll? Won’t our odds increase? Yes! Phenomenally, in comparison. Let’s examine…

Going for Yahtzee on Second Roll

The odds of rolling a Yahtzee on the second roll depend entirely on the production of your first roll. Starting with 4 matching dice on the first roll will significantly increase your odds, of course. To start, we have to calculate the odds of rolling four numbers that match. Let’s pretend they are 5s. Each 5 has a 1/6 chance of rolling, and you need four of those, plus a non-5 that has a 5/6 chance of rolling. The equation looks like this:

(1/6) x (1/6) x (1/6) x (1/6) x (5/6) = 5/7,776

But that’s not all… any one of those five dice could be the dice that didn’t match, so you multiply that 5 by 5 more to get 25/7,776 ways to roll a 4 of a Kind in 5s, or a 0.32% chance. From here, you need that last die to roll a 5 too. There’s a 1/6 chance of that happening on the next roll, which gives us this formula…

(25/7,776) x (1/6) = 25/46,656 = 5.35 (0.053%)

Now, for the final step… the odds of rolling Yahtzee in this way with any number, not just 5. Since there are 6 numbers, we multiply by 5.35 by 6 to get the true probability of 0.32%.

But what if we start with just 3 matching numbers? Or just 2? Or worse yet, no matching numbers at all?! What if we develop these patterns on the second roll, having just a single chance left to roll that oh-so-beautiful Yahtzee?

I could spend hours teaching you these methods and formulas, but I’m not going to because it’s already been done. ThoughtCo explains it all here. If you’re following along up to this point, and have a basic understanding of how to calculate combinations [C(n,r) = n! / (r!(n−r)!)], by all means head over to that page and expand your brain in the ways of Yahtzee math!

If, however, you just want to know the odds of rolling a Yahtzee or any other point-worthy combination, sit tight and keep on reading…

Yahtzee Dice Combination Odds

Due to the multifarious variables that could occur leading up to the final roll, probabilities are listed only in terms of a single roll of all 5 dice. For example, there are far too many starting rolls that could lead to a Large Straight to list them here before you and I both fall asleep trying to understand or calculate them all.

Dice combinations & the odds of rolling them in one toss of all 5 dice are…

Any Pair = 60.19%

Any 3 of a Kind = 15.43%

Any 4 of a Kind = 1.93%

Any Full House = 3.86%

Any Small Straight = 12.35%

Any Large Straight = 3.09%

Any Yahtzee = 0.08%

Using this information and the knowledge you acquired above, you should have an idea of how to calculate your odds of rolling one of these combinations on the second or third roll, depending on how close you are to it on the first or second roll. If, for example, you already have a 3 of a Kind, and are hoping for a Full House, your odds of rolling two dice and getting a pair would be:

1/6 x 1/6 x 6 = 16.66%

Or, the odds that it could turn into a 4 of a Kind would be 33%, since you only need one of those two dice to roll the appropriate number [2/6 = 0.33].

That’s not a bad option when you have the 3 of a Kind, 4 of a Kind, and/or the Upper Section of that number available to score on. If all of those are taken, though, and you must land the Full House to score anything, then 16.66% doesn’t look as pretty.

This is what makes Yahtzee such a deeply scientific and mathematical game. So many variables must be weighed – so many possibilities across so many dice, so many rolls, and a progressively constrictive range to score them in. Every time you touch the dice, your odds of scoring change dramatically, for better or worse.

If the science and math of Yahtzee weren’t fascinating enough, you can learn a lot more about the game in the following pages:

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