Yahtzee is a game of both chance and strategy (strategy is covered in a different article), with a heavy emphasis on the chance. On any givn turn, a player starts by rolling all five dice. Then the player may or may not choose to roll 1 to all 5 of the dice a second time. Following that, the player may choose to roll or not roll 1 or more of the dice a third time. A Yahtzee results any time the previous choices provide the same score on each of the five dice -- for example, 5 fours.
According to my math friend, we use something called the combinatorics formula for combinations to calculate the odds of a Yahtzee and we also need to "break the problem down into a number of mutually exclusive cases." Sounds simple! For him.
So my buddy started with the easiest. For him.
THE ODDS OF GETTING A YAHTZEE ON THE FIRST ROLL
For this example, we'll calculate the chances of getting the five 4's mentioned above. The probability of rolling a 4 on one die is 1/6 (one in six). Makes sense since a die has six sides and the 4 is only on one of those sides.
The formula my math muncher used looks vaguely familiar, so it's probably something I learned in school and promptly forgot -- like cursive writing, playing the recorder, and how to play nicely with others. But I digress. The formula is (1/6) x (1/6) x (1/6) x (1/6) x (1/6) = 1/7776. This reflects the 1 in 6 odds multiplied by the five dice.
But 1/7776 is the probability of rolling a Yahtzee with one specific number (the four's, in our example). Since we would be happy with ANY of the six numbers showing up on all five dice, we have to multiply our first result by six. 6 x 1/7776 = 1/1296. So...
The probability of rolling a Yahtzee on the first roll is 1 in 1296 or 0.08%
THE ODDS OF GETTING A YAHTZEE IN TWO ROLLS
Most likely, you're not going to get that cherished 5-in-a-row on the first toss. That means you'll be rolling at least twice (assuming your goal is a Yahtzee). Adding a second roll makes things a quinterfrabillion times more complicated!
For example, on the first roll you might get four 4's. Then you'll re-roll the one die that doesn't match. Let's assume you get a Yahtzee on this second roll. Here's the math behind those particular two rolls.
One the first roll you get four 4's. Since there was four 1/6 chances of rolling a 4, and one 5/6 chance of not rolling a 4, the formula is (1 x 6) x (1 x 6) x (1 x 6) x (1 x 6) x (5/6) = 5/7776.
But wait a minute! ANY of the five dice could have been the non-4. To express that, the combination formula is C(5,1) =5. You need trust me on that... because I needed to trust my math magician on that.
So if we multiply the first odds by the second odds -- 5/7776 x 5/1 -- we get 24/7776 as the odds of rolling four 4's on the first roll.
Now... we need to calculate the chances of rolling one 4 on one die on our second roll. That would be 1/6. Thus (and I'm only using 'thus' to make it sound like I know what I'm talking about) the chance of rolling a Yahtzee of five 4's in two rolls, assuming the first roll gave us four 4's, is (1/6) x (25/7776) = 25/46656.
But wait! There's more!
There are six different number on each die, so we have to multiply the above by six to reflect that. Hence (I already used 'thus'), 6 x 25/46656 = 150/46656 (or 0.32%).
But wait again! There's more again!
Obviously the above isn't the only way to get a Yahtzee in just two rolls. We could get three of a kind and then match the remaining two dice on the second roll. The chance of that are 6 x C(5,3) x (25/7776) x (1/36) = 0.54%. Or maybe we only match two on the first roll and the remaining three on the second roll. That gives us 6 x C(5,2) x (100/7776) x (1/216) = 0.36 %. Finally, the first roll could give us five non-matching values, so we keep one and roll the remaining four. Assuming that gets us a Yahtzee, the formula would be (6!/7776) x (1/1296) = 0.33%. The '!' in the formula means the factorial for that number. The factorial for 6 is 6 x 5 x 4 x 3 x 2 x 1.
The above cases are mutually exclusive -- they don't rely on each other to get their possible results -- so the possibility of getting a Yahtzee in two rolls is obtained by adding them all up. So...
The probability of rolling a Yahtzee in two rolls is 1.23%
THE ODDS OF GETTING A YAHTZEE IN THREE ROLLS
Go take a couple of aspirin or ibuprofen, and then read the next part.
Obviously, when we add a third roll there are a lot more variations that are possible. We need to calculate the odds of all of them.
The chances of rolling four of a kind, then no match on the second roll, and matching the last die on the third roll is 6 x C(5,4) x (5/7776) x (5/6) x (1/6) = 0.27%.
The odds of rolling three of a kind, then no match, then a matching pair on the last roll is 6 x C(5,3) x (25/7776) x (25/36) x (1/36) = 0.37%.
The probability of getting a matching pair, then zip, then matching the remaining three on the third roll is 6 x C(5,2) x (100/7776) x (125/216) x (1/216) = 0.21%.
The probability of rolling a single die, then absolutely nothing matching it, then getting four of a kind on the third roll is (6!/7776) x (625/1296) x (1/1296) = 0.003%. In other words, not great.
The probability of rolling three matching numbers, matching one more on the next roll, followed by matching the final die on the third roll is 6 x C(5,3) x (25/7776) x C(2,1) x (5/36) x (1/6) = 0.89%.
The odds of rolling a pair, matching an additional pair on the next roll, followed by matching the fifth die on the third roll is 6 x C(5, 2) x (100/7776) x C(3, 2) x (5/216) x (1/6) = 0.89%.
The possibility of rolling a pair, then matching one die on the second roll, followed by matching the last two dice on the last roll is 6 x C(5,2) x (100/7776) x C(3,1) x (25/216) x (1/36) = 0.74%.
The chances of you rolling one of a kind, another die like it on the next roll, and then the needed three of a kind on the final roll is (6!/7776) x C(4,1) x (100/1296) x (1/216) = 0.01%.
The probability of rolling nothing that matches, then three of a kind to match the one you chose on the first roll, followed by a final match on the third roll is (6!/7776) x C(4,3) x (5/1296) x (1/6) = 0.02%.
The odds of rolling one of a kind, then a matching pair on the second roll, and yet another matching pair to match on the last roll is (6!/7776) x C(4,2) x (25/1296) x (1/36) = 0.03%.
When you add up all of the above, it means...
The probability of rolling a Yahtzee in three rolls is 3.43%
THE TOTAL PROBABILITY OF GETTING A YAHTZEE
Now... adding it all up. The chances of getting a Yahtzee in one roll is 0.08%. The odds of doing it in two rolls is much better at 1.23%. And the probability moves up to 3.43% in three rolls. These are mutually exclusive (not dependent on each other) so we add them up to find out that...
The probability of getting a Yahtzee in any given turn is 4.74%
That means, on average, you should get a Yahtzee every 21 or 22 turns, if all you're doing is trying to get that magic 5-in-a-row. However, in a real game of Yahtzee, you'll sacrifice some of those attempts to score in other categories, such as a full house or straight.
