Dice Roll Expected Value Calculator
ANA›Life Services Authority›National Calculator Authority›Dice Roll Expected Value Calculator
.calc-container { max-width: 640px; margin: 2rem 0; padding: 1.5rem; background: #fff; border: 1px solid #ddd; border-radius: 8px; box-shadow: 0 1px 3px rgba(0,0,0,0.06); font-family: system-ui, -apple-system, sans-serif; } .calc-container h3 { font-family: Georgia, serif; font-size: 1.15rem; color: #1a1a1a; margin-bottom: 1rem; padding-bottom: 0.5rem; border-bottom: 2px solid var(--ac, #3d5a80); } .calc-row { display: flex; align-items: center; gap: 0.75rem; margin-bottom: 0.75rem; flex-wrap: wrap; } .calc-row label { min-width: 160px; font-size: 0.9rem; color: #333; font-weight: 500; } .calc-row input[type="number"], .calc-row select { flex: 1; min-width: 120px; max-width: 200px; padding: 0.5rem 0.6rem; border: 1px solid #ccc; border-radius: 4px; font-size: 0.9rem; font-family: system-ui, sans-serif; color: #1a1a1a; background: #fafaf8; } .calc-row input:focus, .calc-row select:focus { outline: none; border-color: var(--ac, #3d5a80); box-shadow: 0 0 0 2px rgba(26,74,138,0.12); } .calc-row .unit { font-size: 0.82rem; color: #888; min-width: 30px; } .calc-btn { display: inline-block; margin-top: 0.5rem; padding: 0.55rem 1.5rem; background: var(--ac, #3d5a80); color: #fff; border: none; border-radius: 4px; font-size: 0.9rem; font-weight: 600; cursor: pointer; font-family: system-ui, sans-serif; } .calc-btn:hover { opacity: 0.9; } .calc-result { margin-top: 1.25rem; padding: 1rem 1.25rem; background: #f0f6fc; border-left: 3px solid var(--ac, #3d5a80); border-radius: 0 6px 6px 0; display: none; } .calc-result.visible { display: block; } .calc-result-label { font-size: 0.78rem; text-transform: uppercase; letter-spacing: 0.06em; color: #666; margin-bottom: 0.25rem; } .calc-result-value { font-size: 1.6rem; font-weight: 700; color: var(--ac, #3d5a80); } .calc-result-detail { font-size: 0.85rem; color: #555; margin-top: 0.5rem; line-height: 1.5; } .calc-note { margin-top: 1rem; font-size: 0.8rem; color: #888; font-style: italic; } .calc-grid { display: grid; grid-template-columns: 1fr 1fr; gap: 0.75rem; margin-top: 0.75rem; } .calc-grid-item { padding: 0.6rem 0.8rem; background: #f8f9fa; border-radius: 4px; border: 1px solid #eee; } .calc-grid-item .label { font-size: 0.75rem; color: #888; text-transform: uppercase; letter-spacing: 0.04em; } .calc-grid-item .value { font-size: 1.1rem; font-weight: 600; color: #1a1a1a; } @media (max-width: 720px) { .calc-row { flex-direction: column; align-items: flex-start; gap: 0.3rem; } .calc-row label { min-width: auto; } .calc-row input[type="number"], .calc-row select { max-width: 100%; width: 100%; } .calc-grid { grid-template-columns: 1fr; } } .calc-chart { margin: 1rem 0; text-align: center; } .calc-chart svg { max-width: 100%; height: auto; } .calc-chart-legend { display: flex; flex-wrap: wrap; justify-content: center; gap: 0.6rem 1.2rem; margin-top: 0.6rem; font-size: 0.8rem; color: #555; } .calc-chart-legend span { display: inline-flex; align-items: center; gap: 0.3rem; } .calc-chart-legend i { display: inline-block; width: 10px; height: 10px; border-radius: 2px; font-style: normal; } .calc-related { max-width: 640px; margin: 2rem 0 1rem; padding: 1.25rem 1.5rem; background: #f8f9fa; border: 1px solid #e8e8e8; border-radius: 8px; } .calc-related h3 { font-family: Georgia, serif; font-size: 1rem; color: #1a1a1a; margin: 0 0 0.75rem; padding-bottom: 0.4rem; border-bottom: 2px solid var(--ac, #3d5a80); } .calc-related-list { list-style: none; padding: 0; margin: 0 0 0.75rem; display: grid; grid-template-columns: 1fr 1fr; gap: 0.4rem 1.5rem; } .calc-related-list li a { font-size: 0.88rem; color: var(--ac, #3d5a80); text-decoration: none; } .calc-related-list li a:hover { text-decoration: underline; } .calc-browse-all { margin: 0.5rem 0 0; font-size: 0.9rem; font-weight: 600; } .calc-browse-all a { color: var(--ac, #3d5a80); text-decoration: none; } .calc-browse-all a:hover { text-decoration: underline; } @media (max-width: 720px) { .calc-related-list { grid-template-columns: 1fr; } }
Dice Roll Expected Value Calculator
Calculate the expected value, variance, and standard deviation when rolling one or more dice with any number of sides.
Number of Dice
Number of Sides per Die
Modifier (added to total roll)
Calculate Results will appear here.
function dicCalc() { var n = parseFloat(document.getElementById('dic-num-dice').value); var s = parseFloat(document.getElementById('dic-num-sides').value); var mod = parseFloat(document.getElementById('dic-modifier').value); var resultDiv = document.getElementById('dic-result');
if (isNaN(n) || isNaN(s) || isNaN(mod)) { resultDiv.innerHTML = 'Please fill in all fields with valid numbers.'; return; } if (!Number.isInteger(n) || n 100) { resultDiv.innerHTML = 'Number of dice must be a whole number between 1 and 100.'; return; } if (!Number.isInteger(s) || s 1000) { resultDiv.innerHTML = 'Number of sides must be a whole number between 2 and 1000.'; return; }
// Expected value of one die: E[X] = (s + 1) / 2 var evOneDie = (s + 1) / 2;
// Expected value of n dice + modifier: E[total] = n * (s+1)/2 + mod var ev = n * evOneDie + mod;
// Variance of one die: Var[X] = (s^2 - 1) / 12 var varOneDie = (s * s - 1) / 12;
// Variance of n dice (independent): Var[total] = n * (s^2 - 1) / 12 var variance = n * varOneDie;
// Standard deviation var stdDev = Math.sqrt(variance);
// Min and max possible roll var minRoll = n * 1 + mod; var maxRoll = n * s + mod;
// Notation var notation = n + 'd' + s + (mod !== 0 ? (mod > 0 ? '+' + mod : mod) : '');
resultDiv.innerHTML = 'Dice Notation: ' + notation + '' + 'Expected Value (Mean): ' + ev.toFixed(4) + '' + 'Variance: ' + variance.toFixed(4) + '' + 'Standard Deviation: ' + stdDev.toFixed(4) + '' + 'Minimum Possible Roll: ' + minRoll + '' + 'Maximum Possible Roll: ' + maxRoll + '' + 'Range: ' + (maxRoll - minRoll); }
#### Formula
For n dice each with s sides and a modifier m:
- Expected Value of one die: E[X] = (s + 1) / 2
- Expected Value of total roll: E[T] = n × (s + 1) / 2 + m
- Variance of one die: Var[X] = (s² − 1) / 12
- Variance of total roll: Var[T] = n × (s² − 1) / 12
- Standard Deviation: σ = √Var[T]
- Minimum Roll: n × 1 + m
- Maximum Roll: n × s + m
Example: 2d6 → E[T] = 2 × (6+1)/2 = 7, Var[T] = 2 × (36−1)/12 ≈ 5.833, σ ≈ 2.415
#### Assumptions & References
More Calculators
- Sprinkler Head Flow Rate & Coverage Calculator
- Snow Removal Time Estimator
- Lighting Control Zone Coverage Calculator
- Sewer Repair Cost Estimator
- Salt & De-Icer Coverage Calculator
- Insulation Resistance Degradation Calculator
- SEER to EER Conversion Calculator — Efficiency Rating Comparison
- Furnace Efficiency Calculator — AFUE to Annual Fuel Cost
- HVAC Filter MERV Rating Selector and Airflow Restriction Estimator
- Static Pressure Drop Calculator for Duct Systems
- HVAC Equipment Rebate Savings Calculator
Read Next
Study Time Planner Authority Network America › Life Services Authority › National Calculator Authority .calc-container { max-width: 640px;...