16 Apr

# Winner’s Edge: Farkle Dice Odds

Gain a competitive edge knowing the odds in Farkle.

Knowing how to play a game gives you a chance to win it. But a chance is only a chance. Knowing the odds of a game – the probabilities of achieving the best result, the best score, the best toss of the dice, as the case may be – that’s how you gain an edge over the competition.

Farkle is one of the most strategy-based dice games you’ll come across; far more intricate than Yahtzee or Qwixx. We’ve already discussed a basic strategy guide for Farkle. Today, we’ll delve deeper into the scientific realm of the game, where mathematical probabilities reign supreme over the Farkle universe!

## Farkle Math & Probabilities

This introduction to 6-dice odds assumes that you are familiar with official Farkle rules and scoring methods. If not, please visit that link for a brief tutorial.

As you know, scoring points requires specific numbers and/or combinations of numbers to be rolled in a single toss of the dice. Any 1 or 5 is worth points. Any 3, 4, 5 or 6 of a Kind scores, as does three Pairs, two 3 of a Kinds, a Pair + 4 of a Kind, or a 1-6 Straight. Below, we’ll discuss the probabilities of each of these combinations resulting from 1 or more rolls of the dice.

A big “Thank you!” to D. Anderson and T. Murphy for catching and alerting us to errors in our math.

## One or Five

The most basic form of scoring is to roll a 1 or 5, worth 100 or 50 points respectively. Since there are 6 possible numbers on the die, and 2 numbers worth rolling, the odds of any single die landing on 1 or 5 is 1 in 3, or 33.33%. We tend to imagine that, with 6 dice to roll, the odds of rolling a 1 or 5 will increase dramatically, and that’s precisely the case. Unlike a coin toss, where every flip has an equal 50/50 chance, the more 6-sided dice you toss, the better your odds of rolling specific numbers – in this case, a 1 or 5.

 Dice Left Possibilities Odds (appx) Probability 6 42,560 of 46,656 10 in 11 91.22% 5 6,752 of 7,776 20 in 23 86.83% 4 1,040 of 1,296 4 in 5 80.25% 3 152 of 216 7 in 10 70.37% 2 20 of 36 5 in 9 55.56% 1 2 of 6 2 in 6 33.33%

## Three of a Kind

The score for each set of dice may vary, but the odds of rolling them do not. The chart below depicts the odds of rolling any 3 of a kind.

 Dice Left Possibilities Odds Probability 6 14,400 of 46,656 1 in 3.24 30.864% 5 1,500 of 7,776 1 in 5.2 19.30% 4 120 of 1,296 1 in 10.8 9.259% 3 6 of 216 1 in 36 2.778% 2 N/A N/A 0.00% 1 N/A N/A 0.00%

## Four of a Kind

Similarly, the odds of rolling any 4 of a Kind are as follows:

 Dice Left Possibilities Odds Probability 6 1,800 of 46,656 1 in 25.92 3.858% 5 150 of 7,776 1 in 51.84 1.929% 4 6 of 1,296 1 in 216 0.463% 3 N/A N/A 0.00% 2 N/A N/A 0.00% 1 N/A N/A 0.00%

## Five of a Kind

And again for a 5 of a Kind…

 Dice Left Possibilities Odds Probability 6 180 of 46,656 1 in 259.2 0.386% 5 6 in 7,776 1 in 1296 0.0772% 4 N/A N/A 0.00% 3 N/A N/A 0.00% 2 N/A N/A 0.00% 1 N/A N/A 0.00%

## Six of a Kind

And finally for 6 of a Kind…

 Dice Left Possibilities Odds Probability 6 6 in 7776 1 in 7776 0.0129% 5 N/A N/A 0.00% 4 N/A N/A 0.00% 3 N/A N/A 0.00% 2 N/A N/A 0.00% 1 N/A N/A 0.00%

## 6-Dice Single Roll Scorers

This section deals with scoring opportunities that can only be thrown in one roll of the dice (i.e. it takes all 6 dice to score). I’m aware that the 6 of a Kind falls into this description, but it seemed better positioned among its ‘# of a Kind‘ cousins above. I’ll toss it in here again for reference.

 6-Dice Combinations Possibilities Odds Probability Straight 1-6 720 of 46,656 1 in 64.8 1.54% Three Pairs 1800 of 46,656 1 in 25.9 3.86% Two Triplets 300 of 46,656 1 in 155 0.64% Full House 450 of 46,656 1 in 103.68 0.965%

## Odds of Farkling

While the odds of banking score per roll are important, most serious players agree it’s more critical to understand how likely you are to Farkle with each consecutive roll of the dice. Regardless of how many points you’ve achieved thus far in the roll or how many dice you have left to toss, knowing the probability of Farkling should always be a major factor in your decision strategy.

 Dice Left Possibilities Odds Probability 6 1,080 of 46,656 1 in 43.2 2.31% 5 600 of 7,776 1 in 13 7.72% 4 204 of 1,296 1 in 6.4 15.74% 3 60 of 216 1 in 3.6 27.78% 2 16 of 36 1 in 2.3 44.44% 1 4 of 6 2 in 3 66.67%

## Odds of Hot Dice

Last but not least, here are the odds of making Hot Dice (scoring with all 6 dice, worthy of another 6-dice throw). It’s more likely than you may think, and the fewer dice you have to throw, the higher your odds of success become. It’s not so advantageous as to impose a positive expectation (1 in 3, or 33.33% at best), but it can be comforting in the final round of play when you’re behind in the count.

 Dice Left Possibilities Odds Probability 6 3,888 of 46,656 1 in 12 8.33% 5 243 of 7,776 1 in 32 3.125% 4 52 of 1,296 1 in 25 4.00% 3 12 of 216 1 in 18 5.55% 2 12 of 36 1 in 9 11.11% 1 2 of 6 1 in 3 33.33%

Interested in learning more about the dice game Farkle? Check out the links below:

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