Farkle is an old dice game popular at parties and in bars. It’s a folk game, so the exact rules and scoring vary from place to place. I was introduced to the game on Facebook, where there is an online flash version. Basically, the game is played in rounds. During each round, you roll up to 6 dice. Certain die combinations are worth points. If you score some points during a roll, then you must take at least some scoring dice away and add those points to your point total for the round. Then you have the option of rolling again or standing pat. If at any time you roll and you don’t score, you “Farkle” and lose all of your points scored so far for the round. If you stand pat, then you can “bank” your points from the current round towards your total points for the game. In the flash version on Facebook, the following die combinations can score:
- Any single 1 or 5
- Three of a kind
- Four of a kind
- Five of a kind
- Six of a kind
- Three pairs
- 1 through 6 straight run
On every successful roll, you should be taking at least one die away. If you happen to score all of your remaining dice during a roll, you can continue the round with all 6 dice again.
So I got to wondering, what are the probabilities of rolling various die combinations in Farkle? How often do you score, and how often do you Farkle? So I made the following tables.
The most important question you should be asking yourself on every throw is, “what are the chances that I’m going to Farkle on this throw?” So we’ll start with a table that shows the odds of Farkling versus various numbers of remaining dice. For each number, a calculated probability is shown with odds. Counting die combinations can be tricky, and so I wrote a small Farkle simulation that would run through 10 million rolls of 6 dice and count the Farkles.
| Number of Dice Left | Probability (Odds) | Number of Farkles Out of 10 Million Throws |
|---|---|---|
| 6 | 2.31% (1 in 43.2) | 231539 |
| 5 | 7.72% (1 in 13.0) | 770832 |
| 4 | 15.74% (1 in 6.4) | 1573927 |
| 3 | 27.78% (1 in 3.6) | 2779829 |
| 2 | 44.44% (1 in 2.3) | 4445297 |
| 1 | 66.67% (2 in 3) | 6664089 |
So the probability of Farkling is pretty low when you’re rolling 6 dice, and it is pretty high (2 in 3) with only one die. But of course we knew that already! More interesting to me is that it is almost 1 in 2 to Farkle on two dice, and almost 1 in 3 to Farkle on three dice!
There are also some special die combinations that you can only get with six dice: the 1 through 6 straight and the triple pairs. Probabilities, odds and simulation results for these cases appear in the next table. I’m also considering the case of double trips – while it does not lead to a special score in the Facebook variation (it is scored just like two “3 of a Kind” throws), it has special value because it clears all the dice.
| Combo | Probability (Odds) | Number of Times Out of 10 Million Throws |
|---|---|---|
| Triple Pairs | 3.86% (1 in 25.9) | 385877 |
| Straight | 1.54% (1 in 64.8) | 154681 |
| Double Trips | 0.64% (1 in 155) | 64182 |
Finally, below is a table which shows the probabilities and odds for getting “N of a Kind” for various numbers of dice being rolled.
| Number of Dice | 6 of a Kind | 5 of a Kind | 4 of a Kind | 3 of a Kind |
|---|---|---|---|---|
| 6 | 0.0129% (1 in 7776) | 0.386% (1 in 259.2) | 4.82% (1 in 20.7) | 30.86% (1 in 3.24) |
| 5 | N/A | 0.0772% (1 in 1296) | 1.93% (1 in 51.8) | 19.3% (1 in 5.2) |
| 4 | N/A | N/A | 0.46% (1 in 216) | 9.26% (1 in 10.8) |
| 3 | N/A | N/A | N/A | 2.78% (1 in 36) |
and the following table shows the simulation results corresponding to the the above table, out of 10 million throws.
| Number of Dice | 6 of a Kind | 5 of a Kind | 4 of a Kind | 3 of a Kind |
|---|---|---|---|---|
| 6 | 1306 | 38439 | 482740 | 3087911 |
| 5 | N/A | 7809 | 192986 | 1929004 |
| 4 | N/A | N/A | 46167 | 926552 |
| 3 | N/A | N/A | N/A | 278109 |
Calculation Details: Probability for Everyone!
In the following section, I will show how to calculate the above Farkle probabilities. But if mathematics makes you run screaming for the door, you’d best turn back now!
But seriously, this stuff isn’t too bad. Calculating probability is all about counting some things that can happen and dividing by everything that can happen. The difficulty in probability is not really in the math, it is in paying attention to how you count things (and making sure you don’t count them twice!) The following table shows the number of possibilities for rolling up to 6 dice.
| Number of Dice | Number of Possibilities |
|---|---|
| 1 | 6 |
| 2 | 36 |
| 3 | 216 |
| 4 | 1296 |
| 5 | 7776 |
| 6 | 46656 |
In the following, I’m going to use the notation (n m) to stand for “the number of combinations of n objects taken m at a time”. Usually the n appears over the m in parentheses without a division bar, and is just n! / (m! (n-m)!) where the “!” denotes the factorial. ( There is a good discussion of these factors at Not About Apples, and it even happens to be in the context of Farkle! ) The table below shows die combinations with interesting scoring patterns where each distinct die result is labeled A through F along with the number of times they show up in the list of all possible die combinations. The number of dice considered in the roll is just the length of the pattern. Sometimes a named pattern like “3 of a Kind” may have more than one letter pattern. For example, with five dice this may be “AAABB” when there is also a pair with the trips or “AAABC” when there is no other pair.
There are two factors (included in the table) which go into counting the number of ways you can get the associated pattern. The first factor is the number of ways in which you can rearrange the dice in the given pattern over the available slots. The second factor is the number of ways you can assign available numbers to the letter variables. The total number of ways you can get the pattern is the product of these two factors.
Care must be taken in evaluating the second parameter: When a particular letter in a pattern is “distinguishable” from the other letters under rearrangement, then the assignment factor just includes the number of available numbers to assign to that letter. When a group of letters in a pattern are “indistinguishable” from each other under rearrangement, then the factor will include (n m) where n is the number of available numbers to assign and m is the number of letters in the letter group. For example, in the pattern “AAABB” (trip A’s and a pair of B’s) the A and the B are distinguishable, because there is no way I can rearrange the A’s and B’s to trick me into thinking I had trip B’s and and a pair of A’s. But in the pattern “AAABC”, the A is distinguishable but the B and the C are indistinguishable, and so for the B and C you want to take the mathematical combination. Essentially, this is because rearranging them gives the same letter pattern, and we already counted all such rearrangements in the first factor!
| Combo | Pattern | Rearrangements | Assignments | Total Number of Ways |
|---|---|---|---|---|
| 6 of a Kind | AAAAAA | (6 6) | 6 | 6 |
| 5 of a Kind | AAAAAB | (6 5)(1 1) | (6)(5) | 180 |
| 5 of a Kind | AAAAA | (5 5) | 6 | 6 |
| 4 of a Kind | AAAABB | (6 4)(2 2) | (6)(5) | 450 |
| 4 of a Kind | AAAABC | (6 4)(2 1)(1 1) | (6)(5 2) | 1800 |
| 4 of a Kind | AAAAB | (5 4)(1 1) | (6)(5) | 150 |
| 4 of a Kind | AAAA | (4 4) | 6 | 6 |
| 3 of a Kind | AAABBC | (6 3)(3 2)(1 1) | (6)(5)(4) | 7200 |
| 3 of a Kind | AAABCD | (6 3)(3 1)(2 1)(1 1) | (6)(5 3) | 7200 |
| 3 of a Kind | AAABB | (5 3)(2 2) | (6)(5) | 300 |
| 3 of a Kind | AAABC | (5 3)(2 1)(1 1) | (6)(5 2) | 1200 |
| 3 of a Kind | AAAB | (4 3)(1 1) | (6)(5) | 120 |
| 3 of a Kind | AAA | (3 3) | 6 | 6 |
| Triple Pair | AABBCC | (6 2)(4 2)(2 2) | (6 3) | 1800 |
| Straight | ABCDEF | (6 1)(5 1)(4 1)(3 1)(2 1)(1 1) | (6 6) | 720 |
| Double Trips | AAABBB | (6 3)(3 3) | (6 2) | 300 |
And following is the Farkle pattern table. Note that since “1″ and “5″ can never appear in a Farkle throw, even though the number of die possibilities in the denomiator is still coming from the above table for six sided dice, the number of assignments will always come out of 4 die possibilities instead of 6!
| Combo | Pattern | Rearrangements | Assignments | Total Number of Ways |
|---|---|---|---|---|
| Farkle | AABBCD | (6 2)(4 2)(2 1)(1 1) | (4 2)(2 2) | 1080 |
| Farkle | AABBC | (5 2)(3 2)(1 1) | (4 2)(2) | 360 |
| Farkle | AABCD | (5 2)(3 1)(2 1)(1 1) | (4)(3 3) | 240 |
| Farkle | AABB | (4 2)(2 2) | (4 2) | 36 |
| Farkle | AABC | (4 2)(2 1)(1 1) | (4)(3 2) | 144 |
| Farkle | ABCD | (4 1)(3 1)(2 1)(1 1) | (4 4) | 24 |
| Farkle | AAB | (3 2)(1 1) | (4)(3) | 36 |
| Farkle | ABC | (3 1)(2 1)(1 1) | (4 3) | 24 |
| Farkle | AA | (2 2) | 4 | 4 |
| Farkle | AB | (2 1)(1 1) | (4 2) | 12 |
| Farkle | A | (1 1) | 4 | 4 |
Most surprising to me here is that there is basically only the one die pattern to Farkle with 6 dice. Cool! So that is why a 6-die Farkle is so rare: you basically have to roll exactly two doubles every time.
Conclusion
These techniques can be changed slightly to cover other dice games like Yahtzee or craps, or also card games. Writing the small Farkle simulation was beneficial because it helped me find a couple of places where I fat-fingered the calculator coming up with the actual numbers. The code used to generate the above numbers is available upon request. I hope you enjoy playing Farkle as much as I do!
(If you liked this post and you’re on LinkedIn, give me a shout!)
(Disclaimer: The figures presented herein are calculated to the best of my knowledge! No warranty is given or implied. If you’re playing for money, use at your own risk! )
Nice post, thanks.
I grew up on the game Zilch, wish is very similar, though I never played the 10 round, simple, version that is presently on Facebook.
As background, I’m an Engineer/Computer guy. Overly analytical and love doing simulations/math modelling. Good times.
So I wrote a simulator myself, that is a fully functional Farkle game (using the rules from FB), wrote in my ’system’ about when to hold ‘em and when to fold ‘em. Making simply the aggressiveness variable. Meaning I won’t bother to keep any thing that is less than x points. 300 by default, up to 1500, etc. Don’t bank unless >= x and then only when you have so many dice left, blah blah.
After debugging I ran a bunch of simulations capturing min/max/avg games. Pretty interesting stuff. I still have more work to do regarding other parts of my system. Meaning right now it always rolls again if it gets three of a kind and has three left. That might be too aggressive if you’re well past the threshold, etc.
Fun stuff. I’m glad I’m not the only one who thinks of simulations for these things…
Thanks again!
Thanks for the comment! It sounds interesting, and I think you’ve hit on a good way to characterize strategy.
If you ever create a web page about your simulation send me a URL and I’ll link to it.
Well, I too just started playing FB Farkle and am now too driven to write a farkle strategy evaluator. Want to evaluate millions of games of farkle with different strategies. Would be interested in knowing how your strategies compare for the FB rules.
I also wrote some code recently to study Farkle odds. Funny how we all did this recently — maybe a surge in interest because of the Facebook app. My family plays the “PocketFarkel” variant in real life, so my numbers are based on that. For the chance of farkling on n dice, the numbers I got match yours very closely, which is great.
If there are only 46656 possible outcomes of rolling 6 dice, why simulate 10 million random rolls? The number of combinations is small enough to enumerate them and get exact counts. Simulating random outcomes as a way of evaluating different strategies does make sense.
Nice observation about the unique pattern to farkle with 6 dice!
You’ve inspired me to blog my own code and results for this. Cheers!
Good point about writing the simulation! The answer is that having just counted the possible outcomes with pencil and paper, I wanted to validate the results with a completely different approach. I was worried about inadvertently duplicating any counting mistakes from the calculation in the program; although in hindsight, the same filters I developed to score the random throws could have been used on a straight up enumeration.
Just want to confirm my Facebook Scoring options for use in my strategy tester…please comment
Facebook Farkle Scoring Possibilities
DieValue QTY Score
1 6 4000
1 5 3000
6 6 2400
1 4 2000
5 6 2000
6 5 1800
4 6 1600
5 5 1500
straight 1 1500
3 6 1200
4 5 1200
6 4 1200
1 3 1000
5 4 1000
3 5 900
2 6 800
4 4 800
3-pair 1 750
2 5 600
3 4 600
6 3 600
5 3 500
2 4 400
4 3 400
3 3 300
1 2 200
2 3 200
1 1 100
5 2 100
5 1 50
Other scoring combos by summing above possibilities
Hmm, didn’t get any notice of replies… I didn’t mean to leave you all hanging!
I should maybe find a place to post the code/app and some screen shots.
I’m a little frustrated in that my strategies aren’t converging as well as I’d expect. Even with simulation 10,000 games, I get more variation in results than I’d expect on repeated runs.
For me, the real interest isn’t about average game score or single max game ever, I think it’s really about % of games over xx. Now is it 9k or 10k? Or higher? Based on my own reactions and to the leaderboard of my friends, 10k is good. Makes me feel good, is reasonably attainable and I can push that last round for something bigger, or buy a round on FB.
So before revealing my strategies, what % of games do you think are over 10k??? :-)
Great post! What language did you program your simulation in?
Thanks!
It was in C#, only a few dozen lines.
10,000 is not nearly enough to accurately simulate the average. Some of the events have very low probability of happening, and you are stringing them along in series, so you are likely missing many combinations (or hitting too many).
Also, when simulating so many random events, you’ll need to do a serious check of your pseudo random number generator (esp. rand()).
Lastly, has the Facebook app reset a couple times this month for y’all, too? My personal best has remained, but the best among friends has reset twice (not just the weekly and monthly). I wonder if they are tweaking the app and some settings do not convert to the new version… And if they are tweaking it, has the scoring changed? I’ve noticed that you cannot take 3-pair if you have 4-of-a-kind and a pair (the 3 pairs are not all unique), even though sometimes it would be a higher score. But I don’t know if you ever could….
I have a question — what is the highest score one can achieve in Face Book Farkle?
Technically it’s unlimited: you could specify any large number, and I could (for example) keep rolling 6 “1’s” over and over again until I exceed it, although it would be more unlikely the higher the number you choose.
You can’t possibly be serious. The results of your simulation are way out. It’s closer to 9% to farkle with 6 dice. Dude, learn statistics, and don’t trust you simulation, even after 10 million runs…
Actually, the result is correct. It is statistically valid because I counted all of the ways you can throw a Farkle with six die (above). And, as someone else already pointed out above, simulating with 10 million throws is already overkill because there are only 46656 6-die combinations and I counted them already. (Although I still contend the simulation was useful from the point of view of attacking the problem cleanly using an alternate method.)
Are we talking about the same figures? I am calculating the odds of Farkling with 6-die on ONE throw (ie- the first throw). Perhaps you are considering the case of rolling up several throws together and failing to reach the 300 point minimum score before Farkling, as in the Facebook variation? I’d agree offhand that seems more like 10%, but that’s not what I calculated.
I have been trying to figure out the odds on Farkling at the end of a frame when you have a one or five with two dice left.
Obviously, you will Farkle 2 out of 3 times throwing one die. Your chart shows you will Farkle 1 out of 2.3 times with two dice.
My question is which would be better in the following scenario. You roll three dice and it comes up 5,5,4. Should you take one 5 or two 5’s? One way you have to roll either a 1 or 5 with one dice, the other way you have to roll either a 1 or 5 twice or either double 1 or 5.
I think it is better statistically to take one 5 at the time but I don’t know enough about the math to prove it.
Whit,
I have always had that same question. So I decided to try and tackle this.
Things to Clear Board:
1. Keep two dice and throw one die. Chance of 1/3 of getting a 1 or 5
2. Keep one die and throw two dice. (Things that must happen)
A. Both dice are 1 or 5 and clears board. Chance 4/36
B. 1st die scores and second does not. Chance 16/32
b. Throw second dice and it scores. Chance 1/3
So we need:
A or (B and b) to clear board
A = 4/36
C = (B and b) = 16/32 * 1/3 = 16/96
I used this formula.
P(A or C) = P(A) + P(C) – P(A and C)
= 4/36 + 16/96 – (4/36 * 16/96)
= 384/3456 + 576/3456 – 64/3456
= 896/3456
= 7/27 or 25.93 % of making it with two dice.
Verses
1/3 = 9/27 or 33.33 % of making it with one die.
I ran a VBA simulation in excel (100 Million iterations) and verified the results.
If someone feels I have errored. Please correct. I’m definately not a Mathematician.
I believe it is better to keep the two dice and roll just one.
Whit:
I generally follow the tenet of “if I haven’t cleared the dice, roll as many as possible.” It’s not always true, but works in general. Let’s investigate your 5,5,4 scenario.
Like you said, keeping the 5s and rolling the 4 only gives you 2 chances to live: 1 and 5. Two good, four bad. 2 of 3 times you’ll farkle.
What about keeping only one 5? Well, that means you’ll have 36 possible rolls, of which 4 clear the dice: 1-5, 5-1, 1-1, 5-5.
However, you can still survive even without clearing the dice with rolls of 1-2, 1-3, 1-4, 1-6, 2-1, 3-1, 4-1, 6-1, 2-5, 3-5, 4-5, 6-5, 5-2, 5-3, 5-4, 5-6.
So, 4 rolls of 36 clear the dice (8-1 odds), 16 of 32 (not counting the 4 that clear) merely keep you alive (1-1) which combine into 20 good rolls for 4-5 odds, better than even money you’ll survive a 2-dice roll.
Of course, I’m talking *real* dice odds- not whatever amusing farce Facebook offers as dice probability.
Based on your odds, I believe the Facebook Farkle is programmed to Farkle at a rate much higher than your information indicates. I will commonly Farkle on 6 dice at least once a game and often time twice.
I’ve been absent, sorry. I’ve run a million+ simulations recently and modified my strategy some based on it. I still don’t have a good place to post the project, or even some pics of the results. I’d be happy to share. .Net/c#. I haven’t been looking at it lately, what with parenting, work, travel, xmas :-).
Mike