# Probability of Yahtzee

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Steven Long
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Joined: 08/08/2013

Have anyone played Yahtzee game?
It's a popular game with 5 dice and you can roll up to 3 times: http://boardgamegeek.com/boardgame/2243/yahtzee

My question relate to this game. Assume that you have 5 dice, in turn you can roll, keep and re-roll any dice up to 3 times.

What is probability in 3 times roll to get:

Specific "2 of a kind": 1-1, 2-2, 3-3, 4-4, 5-5, 6-6.
Specific "3 of a kind":1-1-1, 2-2-2, 3-3-3, 4-4-4, 5-5-5, 6-6-6.
Specific "4 of a kind":1-1-1-1, 2-2-2-2, 3-3-3-3, 4-4-4-4, 5-5-5-5, 6-6-6-6.
Specific "3 in a row": 1-2-3, 2-3-4, 3-4-5, 4-5-6.
Specific "4 in a row":1-2-3-4, 2-3-4-5, 3-4-5-6.
Specific "5 in a row":1-2-3-4-5, 2-3-4-5-6.
Total above or equal 15.
Total above or equal 20.
Total above or equal 25.
Any two different pairs (ex,2-2 and 3-3).
Any full house (ex, 2-2-2-3-3).
Any three of a kind (ex, 4-4-4).
Any four of a kind (ex, 5-5-5-5).
Any small straight (ex, 1-2-3-4).
Any large Straight (ex, 1-2-3-4-5).

Thank you,

saiyanslayer
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Joined: 10/08/2013
Random Math

I've recently read The Art of Game Design: A Book of Lens and highly recommend it. It covers how to calculate those odds, but I'll try my best to help.

Let's look at the odds of rolling a '1' on three dice
( 1 means the specific die was rolled, 0 means it was missed)
1 1 1 (three '1's)
0 1 1 (two '1's and 2-6 on the last die)
1 0 1
1 1 0
0 0 1
0 1 0
1 0 0
0 0 0

You calculate the odds for each of the results above by multiplying the odds for each die.
1 1 1 is three '1's. Rolling a '1' is 1/6. 1/6 x 1/6 x 1/6 = 1/216 (or 1/6 ^ 3). You add each other result to this.
0 1 1 is two '1's and one 2-6. Rolling 2-6 is 5/6 odds. The result would be 5/6 x 1/6 x 1/6 = 5/216
Interesting note here: 0 1 1, 1 0 1, and 1 1 0 all have the same odds. I this case, you just add the results (5/216) together. 5/216 + 5/216 + 5/216 = 15/216.

Ignore the rest of the rolls since they do not have two or more '1's

Add the first result (for the three '1's) together with the other rolls (15/216) for the final tally. 1/216 + 15/216 = 16/216= 2/27 = 7.04%

Let's do it again for four dice, looking for two-of-a-kind.
(I'm leaving out any results that do not matter)
1 1 1 1 = 1/6 x 1/6 x 1/6 x 1/6 = 1/1296
0 1 1 1 = 5/6 x 1/6 x 1/6 x 1/6 = 5/1296 x 4 = 20/1296
1 0 1 1
1 1 0 1
1 1 1 0
1 1 0 0 = 1/6 x 1/6 x 5/6 x 5/6 = 25/1296 x 4 = 100/1296
0 1 1 0
0 0 1 1
1 0 0 1

(4 1's) + (3 1') + (2 1's) = Total Ones
1/1296 + 20/1296 + 100/1296 = 121/1296 = 9.33%

Now to include rerolling? I'm going to have to guess on that one.

Odds of rolling '1' on a d6, with three rerolls: Count each roll as three dice, but only counting if the results come up with one '1' since you'd stop rolling at that point.

1 0 0
0 1 0
0 0 1 = 1/6 x 5/6 x 5/6 = 25/216 x 3 = 75/216 = 34.7%

Each die now has a 75/216 chance to hit a 1 instead of 1/6. Each miss now counts as 141/216 instead of 5/6.

Example: two hits on three dice with three rerolls.
1 1 1 = 75/216 x 75/216 x 75/216 = 421,875/10,077,696 = 4.19%
0 1 1
1 0 1
1 1 0 = 141/216 x 75/216 x 75/216 = 793,125/10,077,696 = 7.87%

If my math is right, that should provide you all the instructions to figure it out. It will take a bit of work though.

Steven Long
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Joined: 08/08/2013
Three dice?

Sorry, but why you give an example three dice? my question is roll on five dice?

Yamahako
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Joined: 12/01/2010
For these, the probability

For these, the probability can get fairly complicated. You really want to calculate the chance of the event in question NOT happening, and then subtract that from one.

You're asking for a significant quantity of work for someone to give you the answers, even for people good at math, but I'll try and explain how to do the first option: Any 2 of a kind.

First off, you will need to understand how many possible results there are. Because each die is an independent event, and has a potential 6 results, to determine the number of possible results you take the number of outcomes of a single element (6 - one for each side on the die), and raise it to the power equal to the concurrent events (5 - one for each die). This would be different if, for example, the die did not have all the same faces or if subsequent dice had different numbers of sides.

So, we can determine that there are 7776 potential outcomes for rolling 5d6. (that's 5, 6 sided die). So how many of them will NOT result in 2 of a kind. Well abstractly, we can see that there are only (6) discrete outcomes that will not end in this result: [1,2,3,4,5]; [1,2,3,4,6]; [1,2,3,5,6]; [1,2,4,5,6]; [1,3,4,5,6]; [2,3,4,5,6]. However, it doesn't matter which of the 5 dice have which result within a particular set's results. If we consider each item in the set to have occurred on its die respectively, then we can determine the potential die sets that could occur by evaluating 5!. "!" is the mathematical operation that is equivalent to X*(X-1)*(X-2)... until the term is 1. 5! would therefore be 120. That means that each set that doesn't result in a pair (for example [1,2,3,4,5]) can occur in 120 different ways. This can be a slightly difficult concept, but you want to think of each die as a discrete value. So in the [1,2,3,4,5] example, the 1 can occur on die A,B,C,D, or E, and each of the subsequent results would obviously need to happen on the other dice.

So we have 6 discrete outcomes that don't result in two of a kind. And each of those results can occur 120 ways. This means there are 720 possible ways out of the total possible 7776 outcomes to NOT get a two of a kind. This gives us a 7056/7776 chance on the FIRST ROLL of getting a two of a kind (90.7%).

Now we get into the other rolls. Because you can choose to re-roll 0-5 dice (6 possible options) we have to now calculate each of the potential results. If you choose to re-roll 0, we know that's a failure (since the first result was a failure).

If you choose to re-roll 1, then you know you have 4 opposite values, so there's only 2 potential values that would give you a failure (because rolling any of the dice you've already rolled would obviously give you a success), so that's 2 outcomes out of 6 that result in failure or 1/3 chance of failure. [6 total potential outcomes]

If you choose to re-roll 2 dice, the first die rolled would have a 1/2 chance of failure (not rolling a double) and the second would have the same (1/3) chance as just rolling 1 die. - when you are looking at cumulative chance within the same event, you multiple the results. So when re-rolling 2 die, you have a 1/6 cumulative chance of failure (for the 2nd roll event). [36 total potential outcomes]

If you choose to re-roll 3 dice, you have a 2/3 chance for the first, 1/2 chance for the second, and a 1/3 chance for the third for a total of 1/9 chance of failure. [216 potential outcomes]

If you choose to re-roll 4 dice, you have a 5/6 chance for the first, 2/3 chance for the second, 1/2 chance for the third, and a 1/3 chance for the fourth. This results in a 10/108 chance of failure. [1296 potential outcomes]

If you re-roll all 5 dice, you have the same result as the original roll, or 720/7776.

So the chance for each result is distinct for each determinant. Because there are 6 potential options (rolling 0-5 dice), we need to look at the total failures out of the complete set of potential outcomes. When rolling 1-5 dice, there are a cumulative 9330 potential new outcomes. This means we have a cumulative quantity of 72550080 outcomes. Let's see the breakdown of the failures:

When re-rolling 0 dice - we know there are only 720 potential failure results.
When re-rolling 1 die - we know there are 720+ an additional 2 failure results. (722)
When re-rolling 2 dice - we know there are 720+ an additional 6 failure results. (726)
When re-rolling 3 dice - we know there are 720+ an additional 24 failure results. (744)
When re-rolling 4 dice - we know there are 720+ an additional 10 failure results. (730)
When re-rolling 5 dice - we know there are 720+ an additional 720 failure results. (1440)

5082 failures in 72550080 outcomes, which means we should see 72544998 successes in 72550080 outcomes, or 99.993% cumulative chance of success. However, there is another potential roll.

Luckily things don't change much with the second roll, the number of results are the same, and the potential failures are the same. But the number of potential results is vastly different. This is because there are 6 options for EACH of the previous options. I'm going to code them A-B-C-D-E-F below, so that BD would be re-rolling 1 die the first time, and 3 dice the second time. This means there are a staggering 47855048216844521080163520000000 potential results...

AA - 720 failures +0 failures (720)
AB - 720 failures +2 failures (722)
AC - 720 failures +6 failures (726)
AD - 720 failures +25 failures (744)
AE - 720 failures +10 failures (730)
AF - 720 failures +720 failures (1440)
BA - 722 failures +0 failures (722)
BB - 722 failures +2 failures (724)
BC - 722 failures +6 failures (728)
BD - 722 failures +24 failures (746)
BE - 722 failures +10 failures (732)
BF - 722 failures +720 failures (1442)
CA - 726 failures +0 failures (726)
CB - 726 failures +2 failures (728)
CC - 726 failures +6 failures (732)
CD - 726 failures +24 failures (750)
CE - 726 failures +10 failures (736)
CF - 726 failures +720 failures (1446)
DA - 744 failures +0 failures (744)
DB - 744 failures +2 failures (746)
DC - 744 failures +6 failures (750)
DD - 744 failures +24 failures (768)
DE - 744 failures +10 failures (754)
DF - 744 failures +720 failures (1464)
EA - 730 failures +0 failures (730)
EB - 730 failures +2 failures (732)
EC - 730 failures +6 failures (736)
ED - 730 failures +24 failures (754)
EE - 730 failures +10 failures (740)
EF - 730 failures +720 failures (1450)
FA - 1440 failures +0 failures (1440)
FB - 1440 failures +2 failures (1442)
FC - 1440 failures +6 failures (1446)
FD - 1440 failures +24 failures (1464)
FE - 1440 failures +10 failures (1450)
FF - 1440 failures +720 failures (2160)

For a total of 35064 failures out of a total of 47855048216844521080163520000000 potential outcomes. So that means we are looking at 47855048216844521080163519964936 success or 99.99999999999999999999999993% chance of success.

Now, when you're playing the game, your individual chances will be different because you would only apply the chance for the particular number of dice you are choosing to reroll. This is the chance of success if all choices are completely random and you stop on success.

That's just ONE of the problems. When you are looking for things that can have multiple values on a single die, it starts getting significantly more complex.

Does that help at all?

drunknmunky
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Joined: 06/25/2010
I'm fairly sure

I'm fairly sure saiyanslayer's math is wrong. The odds of rolling a single 1 on 3 dice with 1 roll is 50% not 7.04%. With the goal of a single 1 you simply add the probability of each die together. 1/6+1/6+1/6=3/6. You're overcomplicating the issue.

Just a quick reference, if your probability percent goes down when adding more dice, you're doing it wrong. More dice without a change to the goal will always increase the percentage.

Two 1s on two dice is a 1/36 chance. On three dice I believe it is 7/36 or 1/6x1/6=1/36(the flat probability of 2 dice giving the correct result) + 6/36(the odds of the spare die giclving the correct result).

Now this does become more complicated when you want an EXACT result. If in the above you only want exactly two 1s, extra die add to the odds of getting too many 1s on a roll and I believe then, your math comes into play.

Steven Long
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Joined: 08/08/2013
My really problem

I will simplify my problem. I ask some guys and they help me calculate this table, it's probability of 5 dice in ONE time roll:

30.52% Sum is 20 or more
27.01% Two different pairs
21.30% Three of a kind
19.62% 1-1, 2-2, 3-3, 4-4, 5-5, 6-6
15.82% 1-2-3, 2-3-4, 3-4-5, 4-5-6
15.43% small straight
15.20% Sum is 22 or more
6.17% 1-2-3-4, 2-3-4-5, 3-4-5-6
5.88% Sum is 24 or more
3.94% Full House (inclusive five of a kind)
3.55% 1-1-1, 2-2-2, 3-3-3, 4-4-4, 5-5-5, 6-6-6
3.09% large straight
2.01% Four of a kind
1.62% Sum is 26 or more
1.54% 1-2-3-4-5, 2-3-4-5-6
0.33% 1-1-1-1, 2-2-2-2, 3-3-3-3, 4-4-4-4, 5-5-5-5, 6-6-6-6
0.27% Sum is 28 or more
0.08% Five of a kind

You can see that order is arranged from easiest to hardest.
My really problem that i don't know this order 'll change on 3 times roll or not?
Thanks!

drunknmunky
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Joined: 06/25/2010
The issue is you're adding a

The issue is you're adding a variable die roll to the probability. Look at it this way. In my example above an extra die adds a 1/6th chance to get a given number. This still works no matter then number of extra dice. So...

.08% for yatzee on the first roll.
.08%+1/6^n where "n" is the number of rerolled dice on second roll
Previous %+1/6^n where "n" is the number of rerolled dice for the third roll

So in each additional roll you're adding to the total percent the percent of rolling all the same number on the rerolled dice.

If you rerolled 3 dice on roll2 you have a 0.47% chance of yatzee. If you reroll 2 dice on roll3 you have a 3.25% chance of yatzee. If you had only rerolled 2 on roll2 you have a 2.79% chance total. If you reroll 1 on roll3 you have a 19.45% chance.

This is just a compounding percentage for getting exact numbers. The percentages are for getting a set 5 numbers across all three rolls in one turn, not an updated percentage per roll. To do that you just take the percent that you quoted above depending on the number of dice you rolled, but that isn't important really

drunknmunky
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Joined: 06/25/2010
On a side note, I didn't

On a side note, I didn't double check your perelcentages. I have no clue where you got 0.08% on one roll for yatzee. Any single roll of 5 dice looking for 5 exact numbers is 1/7776 which is .012%. Additional rolls adjust that as above, but the base is always the same.

Am I missing something there?

Edit:
I'm an idiot. I forgot there are 6 cases of yatzee in one roll and 6 times .012 is .072 or about .08

This actually skews my above example percent as I forgot to take it into account, but only by like .06% each so not enough to really change the point.

Yamahako
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Joined: 12/01/2010
Do you mean at LEAST a result

Do you mean at LEAST a result of [1] or ONLY a result of [1]. At least a result of one would be 1/2 (50%) (1/6 + 1/6 + 1/6) you are correct. A result of [1] ONLY would be (1/6)*(5/6)*(5/6) or 25/216 (11.6%).

A result of [1,1,ANY] on 2d6 is 1/6 (16.7%) not 7/36. You can roll [1,1,ANY] which there are 6 of, but you can roll those results 6 different ways (3!) for 36 potential results out of 216 (6^3) potential outcomes, which reduces to 1/6. For a result of [1,1,NOT1] you have 5 potential results and 6 potential ways to roll those results (3!) for 30 potential results out of 216 (6^3) options, which reduces to 5/36 (13.9%).

Bruce Long, the amount of work you want done for you is staggering. While it would stand to reason that incorporating the multiple rolls, the chances order would remain similar, it isn't necessarily true and would take a lot of calculation to be sure.

Why this is the case, is because once you're dealing with things that depend on many particular outcomes (like full houses, for example), there may be ways to specifically improve your chances by choosing specific dice to re-roll.

For example, in the two of a kind example, we can see that it is best to only re-roll 4 or 5 dice for the best chance of success. What the correct option is, might not always be obvious.

If there's a part of the problem you don't understand, I'll be glad to help you understand it so that you can do the calculations yourself, but I don't have the time to figure out the 36 probability trees for each of those options to give discrete chances. This isn't difficult arithmetic, the calculator on your computer can do it all, its the logic that can take some time on - but experience doing it is the best way to learn.

*EDIT* Fixed some math

drunknmunky
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Joined: 06/25/2010
Oh yea! You're right there. I

Oh yea! You're right there. I was not factoring in getting ONLY the wanted amount. That does screw with the math. Thank you. Thought I was missing something.

StagCutlery
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Joined: 06/02/2013
It's not a new game, so there're websites for it.

Don't know if you searched on the web yet:

http://www.datagenetics.com/blog/january42012/

Steven Long
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Joined: 08/08/2013
Yamahako wrote:Bruce Long,

Yamahako wrote:

Bruce Long, the amount of work you want done for you is staggering.

Sorry, i don't know it too complicate such that.

Yamahako wrote:

While it would stand to reason that incorporating the multiple rolls, the chances order would remain similar, it isn't necessarily true and would take a lot of calculation to be sure.

So the answer that It'll no change order of probability in three roll, right?

Yamahako
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Joined: 12/01/2010
I can't be certain. Two of

I can't be certain. Two of the options (2 pairs, and full house) have some complex interaction with them. Which dice you choose to keep change the outcome for these dramatically. It's possible that when incorporating multiple rolls, they could jump above the options around it.

Steven Long
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Joined: 08/08/2013
order

Thanks, i just want to know their order, to arrange points according hard level. Perharps playtest 'll resolve.

Steven Long
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Joined: 08/08/2013
One question more:

One question more:
What's the probability of "All odd numbers on 5 dice" (1,1,3,3,5) and "All even numbers on 5 dice" (2,4,4,6) in one roll?
Thanks!

X3M
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Joined: 10/28/2013
Yamahako wrote:Do you mean at

Yamahako wrote:
Do you mean at LEAST a result of [1] or ONLY a result of [1]. At least a result of one would be 1/2 (50%) (1/6 + 1/6 + 1/6) you are correct. A result of [1] ONLY would be (1/6)*(5/6)*(5/6) or 25/216 (11.6%).

Maybe I am missing something here, but...

Are you talking about rolling 3 dice in one go?
And you want to throw at least one 1? If so, then the "at least a result of [1]" calculation is incorrect.
Imagine throwing 6 dice. You think the chance of having at least a result of [1] is 100% then? I disagree on that.

At least a result of [1], so the three dice produce one, two or three [1]:
First dice has 1/6 or 36/216
Second dice has 5 * 1/36 = 5/36 or 30/216
Third dice has 25* 1/216 = 25/216
Adding up: 91/216 (42,1%, not 50%)

The other one isn't complete either. A result of one [1] only, so the three dice produce [1][not 1][not 1] and this [1] can be switch with a [not 1], so times 3, would be:
(1/6)*(5/6)*(5/6) + (5/6)*(1/6)*(5/6) + (5/6)*(5/6)*(1/6) = 75/216 (34,7%, not 11,6%)

But then again.
My apologies when I am missing the point.