Farkle Probabilities
Following are the probabilities of scoring combinations in the initial throw of six dice.
| Dice combination | Probability |
|---|---|
| Three of a kind | 1 in 3.240 |
| Four of a kind | 1 in 20.736 |
| Three pair | 1 in 25.92 |
| Straight | 1 in 64.8 |
| Two triplets | 1 in 155.52 |
| Five of a kind | 1 in 259.2 |
| Six of a kind | 1 in 7776 |
For the most part, less probable combinations are scored higher than more probable combinations (see Scoring Variations).
Following are the probabilities of scoring combinations in subsequent throws of the dice.
| Dice combination | Five dice left | Four dice left | Three dice left |
|---|---|---|---|
| Three of a kind | 1 in 5.184 | 1 in 10.8 | 1 in 36 |
| Four of a kind | 1 in 51.84 | 1 in 216 | N/A |
| Five of a kind | 1 in 1296 | N/A | N/A |
Following are the probabilities of farkling if all variant scoring combinations are allowed, depending on the number of dice thrown.
| Dice thrown | Probability |
|---|---|
| 6 | 1 in 42 |
| 5 | 1 in 13 |
| 4 | 1 in 6.35 (20 in 127) |
| 3 | 1 in 3.6 (5 in 18) |
| 2 | 1 in 2.25 (4 in 9) |
| 1 | 1 in 1.5 (2 in 3) |
Three-pair is the only scoring variation that alters the likelihood of farkling, and only on the initial throw of six dice. If three pairs are not scored, the probability of farkling on the initial throw increases to 1 in 32.
Following are the probabilities of making hot dice in a single throw if all variant scoring combinations are allowed, depending on the number of dice thrown.
| Dice thrown | Probability |
|---|---|
| 6 | 1 in 12 |
| 5 | 1 in 32 |
| 4 | 1 in 25 |
| 3 | 1 in 18 |
| 2 | 1 in 9 |
| 1 | 1 in 3 |
If no variant scoring combinations are allowed, the probabilities of making hot dice are decreased only slightly for 4-6 dice thrown, and unchanged for 1-3 dice thrown. Odds for these and other die combinations with explanations and simulation results can be found elsewhere.
Read more about this topic: Greed (dice Game)