Yatzy Dice Probability Charts — Complete Mathematical Reference

Understanding Yatzy probability transforms you from a casual player into a strategic one. Instead of going with your gut, you can make mathematically optimal decisions about which dice to keep and which categories to pursue. This guide provides complete probability charts for every Yatzy situation.

Basic Dice Mathematics

Before diving into specific combinations, let’s establish the fundamentals.

Total Possible Outcomes

When you roll 5 six-sided dice:

• Each die has 6 possible outcomes
• Total combinations: 6 × 6 × 6 × 6 × 6 = 7,776

This means each specific roll (like 1-2-3-4-5 in exact positions) has a 1/7,776 chance of occurring.

Expected Value Per Die

• Average value of one die: (1+2+3+4+5+6) ÷ 6 = 3.5
• Expected total of 5 dice: 3.5 × 5 = 17.5

This helps you evaluate Chance (in Yahtzee) and understand if a roll is above or below average.

Single-Roll Probabilities

These are the odds of achieving each combination on exactly one roll of five dice, with no rerolls.

Upper Section Numbers

Key insight: On any single roll, you’ll average less than one of any specific number. This is why the upper section bonus (requiring 63+ points, or averaging three per category) is challenging.

Lower Section Combinations

Multi-Roll Probabilities

Yatzy gives you up to three rolls per turn. These charts show your odds of achieving combinations when you play optimally (keeping the best dice and rerolling the rest).

Yatzy (Five of a Kind) Probabilities

The probability across a full turn assumes you start by rolling all five dice and make optimal keepers.

Large Straight (2-3-4-5-6) Probabilities

Full House Probabilities

Three of a Kind Probabilities

Probability of Matching Dice

How likely are you to roll certain patterns? This table shows combinations, not permutations (so 1-1-2-3-4 and 2-1-1-3-4 count as the same pattern).

Expected Scores by Category

These are the average scores you’d expect if you dedicated a full turn to each category, making optimal decisions.

Upper Section Expected Scores

Notice that the expected upper section score (~52.5 points) falls short of the 63-point bonus threshold. This means you need above-average luck or strategic play to earn the bonus. Most successful players:

• Target higher numbers more aggressively
• Accept slightly below-target scores on ones and twos
• Know when to take a zero on a low category to protect higher ones

Lower Section Expected Scores

The “including zeroes” note is important — these expected values account for turns where you fail to achieve the category and must score zero.

Conditional Probabilities: What to Do After Roll 1

You Have a Pair — What Are Your Odds?

You Have Three of a Kind — What Are Your Odds?

You Have Four of a Kind — What Are Your Odds?

You Have 4/5 of a Straight — What Are Your Odds?

If you need exactly one specific number to complete a straight:

If you’re missing one number that could be two values (like needing 1 OR 6 for a 4-sequence):

Strategic Probability Applications

The “Break-Even” Points

When should you pursue a risky category instead of a safe one?

Going for Yatzy vs. Four of a Kind:

• Four of a Kind with sixes: 24 points (guaranteed if you have it)
• Yatzy: 50 points × 4.6% probability ≈ 2.3 expected points

If you already have four of a kind, the choice is:

• Take 24 guaranteed points, or
• Roll for Yatzy (16.7% success): 50 × 0.167 ≈ 8.3 expected points

In this case, taking the guaranteed Four of a Kind (24 points) is better than the expected value of going for Yatzy (8.3 points) unless you’ve already filled Four of a Kind.

Full House vs. Three of a Kind: If you have three of a kind and need to choose between trying for Full House (worth ~25 points with high dice) or taking Three of a Kind (worth ~18 points max):

• Probability of Full House from Three of a Kind: ~33%
• Expected Full House value: 0.33 × 25 = 8.25 points (plus 67% chance of zero)
• Three of a Kind now: 15-18 points guaranteed

Take the Three of a Kind unless you’ve already filled it.

Upper Section Bonus Mathematics

The 63-point bonus threshold requires an average of 3× each number across all six categories:

• Three 1s = 3 points
• Three 2s = 6 points
• Three 3s = 9 points
• Three 4s = 12 points
• Three 5s = 15 points
• Three 6s = 18 points

Total: 63 points exactly. But you get three rolls per turn, and the expected value per die is 2.5 of any specific number. This means:

• Probability of achieving bonus naturally: ~35%
• Probability if you prioritize upper section: ~65%

The 50-point bonus (in Yatzy) divided by six categories means each upper section category is worth approximately 8.3 “bonus points.” Keep this in mind when deciding whether to sacrifice a lower upper-section score.

Probability Comparison: Yatzy vs. Yahtzee

Some probabilities differ slightly due to rule differences:

Yahtzee players achieve straights far more often because they only need four (not five) consecutive numbers for the Small Straight.

Quick Reference Probability Table

Print this or bookmark it for quick lookup during games:

Conclusion

Yatzy probability isn’t about memorizing every number — it’s about understanding the relative likelihood of each combination. The key takeaways:

1. Straights are rare (0.46% per roll), so prioritize them when you see potential
2. Yatzy is very rare (0.08% per roll), but not impossible with focused effort (~5% per turn)
3. The upper section bonus is hard — you need to actively protect those categories
4. Expected values trump hope — a guaranteed 20 points beats a 10% chance at 50 points

Use these probability charts to inform your decisions, but remember that Yatzy is still a game of chance. The math helps you make the best decision more often, but the dice will surprise you both ways. That’s what makes the game exciting.