Cribbage Math: Probabilities, Statistics & Expected Values
The complete mathematical guide to cribbage. Hand probabilities, expected values for keeps and discards, pegging statistics, and the numbers that help you make optimal decisions.
Cribbage Math: Probabilities, Statistics & Expected Values
Behind every cribbage decision lies mathematics. Understanding the numbers won’t guarantee wins, but it transforms intuition into informed strategy. This guide covers the key statistics every serious cribbage player should know.
Hand Score Distribution
Average Values
| Metric | Hand | Crib |
|---|---|---|
| Mean (average) | 7.8 points | 4.7 points |
| Median | 8 points | 4 points |
| Mode (most common) | 4 points | 0 points |
| Standard deviation | 4.2 points | 3.9 points |
Score Frequency Table
How often each score appears (across all possible hands):
| Score | Frequency | Score | Frequency |
|---|---|---|---|
| 0 | 1.6% | 13 | 2.3% |
| 1 | 0.8% | 14 | 3.0% |
| 2 | 7.2% | 15 | 1.5% |
| 3 | 2.8% | 16 | 2.9% |
| 4 | 11.1% | 17 | 1.0% |
| 5 | 5.4% | 18 | 0.7% |
| 6 | 9.8% | 19 | 0% ★ |
| 7 | 6.8% | 20 | 0.88% |
| 8 | 10.8% | 21 | 0.51% |
| 9 | 6.3% | 22 | 0.16% |
| 10 | 7.1% | 23 | 0.10% |
| 11 | 3.4% | 24 | 0.34% |
| 12 | 8.4% | 25-27 | 0% ★ |
| 28-29 | <0.01% |
★ Mathematical impossibilities
Impossible Scores
These point totals cannot be achieved with any 5-card combination:
- 19 points — The famous “nineteen hand” (actually 0)
- 25 points — Gap between 24 and 28
- 26 points — No valid combination exists
- 27 points — No valid combination exists
Why do these gaps exist? The scoring rules (multiples of 2 for fifteens, specific run structures) create mathematical constraints that skip these values.
The Value of Individual Cards
Card Contribution to Hand Value
The average contribution each card makes when part of a 5-card scoring combination:
| Card | Avg Contribution | Notes |
|---|---|---|
| 5 | 4.6 points | Most valuable—combines with 10s for 15 |
| J | 3.4 points | Nobs potential + 10-value for 15s |
| 10, Q, K | 3.1 points | 10-value for 15s |
| 4 | 2.9 points | Good for runs, makes 15 with J |
| 6 | 2.8 points | Runs, 15 with 9 |
| 7 | 2.7 points | Runs, 15 with 8 |
| 3 | 2.5 points | Runs, flexible |
| 8 | 2.5 points | 15 with 7 |
| 9 | 2.4 points | 15 with 6 |
| 2 | 2.3 points | Low runs, limited 15s |
| A | 1.8 points | Fewest combinations |
Key insight: 5s are dramatically more valuable than any other card.
The 5 Advantage
The 5 combines with 16 cards (all 10-value cards) to make fifteens. This is why:
- Never throw 5s to opponent’s crib
- 5-5 is an excellent crib contribution
- Holding a 5 significantly increases expected hand value
Discard Expected Values
Best Two-Card Combinations for Your Crib
| Discard | Expected Crib Points |
|---|---|
| 5-5 | 8.5+ |
| 5-10/J/Q/K | 6.0-6.5 |
| Pair (non-5) | 4.5-5.5 |
| 5-6 or 5-4 | 5.0-5.5 |
| Suited connector | 4.0-4.5 |
| Touching cards | 3.5-4.0 |
| Random cards | 2.0-3.0 |
Worst Two-Card Combinations for Opponent’s Crib
| Discard | Avg Points Given |
|---|---|
| A-K (far apart) | 2.0 |
| 2-9, A-8 | 2.2-2.5 |
| Non-touching, non-5 | 2.5-3.0 |
| Touching cards | 4.0+ |
| Cards totaling 5 | 4.5+ |
| Cards totaling 15 | 5.0+ |
| Any 5 | 5.5+ |
Cut Card Probabilities
Impact of Starter Card on Hand Value
The starter card affects your hand’s expected value:
| Your Hand Profile | Avg Improvement from Cut |
|---|---|
| Already has runs | +1.5 points (extend potential) |
| Has pair | +2.0 points (pair royal chance) |
| Has three-of-a-kind | +3.0 points (four-of-a-kind chance) |
| Has 4-card flush | +1.5 points (5th suit chance) |
| Many 5-card combos | +2.5 points (multiple chances) |
| Low synergy hand | +1.0 points (limited improvement) |
Probability of Helpful Cuts
When counting potential improvements:
| Looking For | Cards Remaining | Probability |
|---|---|---|
| Specific card (e.g., one 5) | 1 | 2.2% |
| Any of 4 cards (e.g., any Jack) | 4 | 8.7% |
| Any of 8 cards (two ranks) | 8 | 17.4% |
| Any 10-value card | 16 | 34.8% |
| Any card helping | 20+ | 43%+ |
Calculation: (Helpful cards) ÷ 46 unknown cards × 100
(You know 6 dealt cards, leaving 46 unknown)
Pegging Statistics
Average Pegging Points Per Hand
| Situation | Dealer | Pone |
|---|---|---|
| Average pegging | 2.8 | 3.5 |
| Skilled defender | 2.2 | 2.8 |
| Aggressive pegging | 3.5 | 4.2 |
Pone pegs more because they lead and have last card advantage.
Common Pegging Scores
| Pegging Event | Probability |
|---|---|
| 15 (exactly) | ~15% of plays |
| 31 (exactly) | ~8% of plays |
| Pair | ~12% of plays |
| Run of 3+ | ~7% of plays |
| Go (1 point) | ~35% of hands end this way |
The Value of Last Card
The “go” or “last card” advantage:
- Guaranteed 1-2 points (go or 31)
- Additional peg opportunities
- Pone averages +0.7 points from last card
First Deal Advantage
Does Going First (Pone) or Dealing Matter?
Over a complete game to 121:
| Metric | First Dealer | First Pone |
|---|---|---|
| Win rate | 49.8% | 50.2% |
| Average winning margin | 117-121 | 117-121 |
The difference is negligible. The deal alternates enough that initial position barely matters.
However, being dealer on specific hands does matter:
- Dealer averages +4.7 (crib) - 0.7 (pone pegging advantage) ≈ +4.0 points per hand
- Getting “last deal” when close to 121 is significant
Game Outcome Probabilities
Expected Points Per Hand (Total)
| Role | Hand | Pegging | Crib | Total |
|---|---|---|---|---|
| Dealer | 7.8 | 2.8 | 4.7 | 15.3 |
| Pone | 7.8 | 3.5 | 0 | 11.3 |
| Difference | 0 | -0.7 | +4.7 | +4.0 |
The dealer has a ~4-point advantage per hand dealt.
Skunk Probabilities
| Outcome | Probability |
|---|---|
| Normal win (opponent 90+) | ~65% |
| Skunk (opponent 61-90) | ~30% |
| Double skunk (opponent <61) | ~5% |
Making EV-Based Decisions
Example: Simple Discard Decision
You’re dealt: 4-5-5-6-9-K
Keep options:
- 5-5-6-9 → Keep value: ~10 points, Cut improvement: moderate
- 4-5-5-6 → Keep value: ~12 points, Cut improvement: run potential
- 5-5-9-K → Keep value: ~8 points, Cut improvement: limited
Your crib discard matters too:
| Keep | Discard | Hand EV | Crib EV | Total EV |
|---|---|---|---|---|
| 4-5-5-6 | 9-K | 12.1 | 3.2 | 15.3 |
| 5-5-6-9 | 4-K | 10.4 | 3.0 | 13.4 |
Optimal: Keep 4-5-5-6, discard 9-K
When to Deviate from EV Calculations
Pure expected value works for average situations. Adjust for:
- Board position — Need to peg when far behind
- Close to 121 — Hand value may not matter
- Skunk danger — Minimize variance
- End game — Specific point needs
Quick Reference: Key Numbers to Remember
Must-Know Statistics
| Fact | Value |
|---|---|
| Average hand score | 7.8 points |
| Average crib score | 4.7 points |
| Dealer’s per-hand advantage | ~4 points |
| Most common score | 4 points |
| Odds of a 29 hand | 1 in 216,580 |
| 10-value cards in deck | 16 |
| Impossible scores | 19, 25, 26, 27 |
Card Values for Quick Decisions
- 5s: Always valuable, never discard to opponent
- Pairs: Worth keeping unless hand is weak
- Touching cards: Run potential adds ~2-3 EV
- Wide spreads (A-K, 2-9): Safe discards to opponent
Advanced: Monte Carlo Analysis
Serious players use computer simulations to evaluate difficult decisions:
- For a given keep/discard, simulate thousands of cuts
- Average the resulting hand scores
- Add estimated crib value
- Compare all 15 keep options
This is how “optimal” discard tables are generated. You don’t need to do this math yourself—just understand that such analysis confirms the standard strategy recommendations.
Putting Math Into Practice
Understanding these numbers helps you:
- Make close decisions — When two keeps seem equal, EV provides guidance
- Evaluate your play — Track your crib averages; are you maximizing?
- Understand variance — A 12-point hand is above average; don’t be greedy
- Recognize opportunity — Dealt a 5? You’re starting ahead.
Want to calculate specific hand values? Use our Cribbage Hand Calculator to explore any combination, or play a live game to see the math translate into real decisions.