likely are you to roll a Yahtzee, or any point-worthy combination, in
one to three rolls of the dice?
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.
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
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.
the formula for…
Dice: 6 x 6 x 6 = 216 possibilities
Dice: 6 x 6 x 6 x 6 = 1,296 possibilities
Dice: 6 x 6 x 6 x 6 x 6 = 7,776 possibilities
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
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.
how many outcomes will give us a Yahtzee? Six of them:
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
the equation is 6/7,776 = 7.716, which rounds to about 0.08%.
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
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:
x (1/6) x (1/6) x (1/6) x (5/6) = 5/7,776
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…
x (1/6) = 25/46,656 = 5.35 (0.053%)
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%.
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
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!
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
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
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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