Card Draw Probability Calculator
ANA›Life Services Authority›National Calculator Authority›Card Draw Probability 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; } }
Card Draw Probability Calculator
Calculate the probability of drawing at least a certain number of desired cards when drawing from a deck (hypergeometric distribution).
Total Cards in Deck (N)
Desired Cards in Deck (K)
Number of Cards Drawn (n)
Desired Successes (k)
Calculate Results will appear here.
function carLogFactorial(n) { // Use Stirling's approximation for large n, exact for small n if (n n) return -Infinity; if (k === 0 || k === n) return 0; return carLogFactorial(n) - carLogFactorial(k) - carLogFactorial(n - k); }
function carHypergeomPMF(N, K, n, k) { // P(X = k) = C(K,k) * C(N-K, n-k) / C(N, n) if (k Math.min(K, n)) return 0; if ((n - k) > (N - K)) return 0; var logP = carLogComb(K, k) + carLogComb(N - K, n - k) - carLogComb(N, n); return Math.exp(logP); }
function carCalc() { var N = parseInt(document.getElementById('car-deck-size').value); var K = parseInt(document.getElementById('car-success-states').value); var n = parseInt(document.getElementById('car-draw-size').value); var k = parseInt(document.getElementById('car-desired-draws').value); var resultDiv = document.getElementById('car-result');
// Validation if (isNaN(N) || isNaN(K) || isNaN(n) || isNaN(k)) { resultDiv.innerHTML = 'Please fill in all fields with valid numbers.'; return; } if (N Deck size must be at least 1.'; return; } if (K N) { resultDiv.innerHTML = 'Desired cards in deck must be between 1 and N.'; return; } if (n N) { resultDiv.innerHTML = 'Cards drawn must be between 1 and N.'; return; } if (k Math.min(K, n)) { resultDiv.innerHTML = 'Desired successes must be between 0 and min(K, n) = ' + Math.min(K, n) + '.'; return; }
// P(X = k) exact var pExact = carHypergeomPMF(N, K, n, k);
// P(X >= k) = sum from i=k to min(K,n) var pAtLeast = 0; var maxK = Math.min(K, n); for (var i = k; i ' + i + '' + pct + '%'; } if (maxK > 14) { tableRows += '... (truncated)'; }
resultDiv.innerHTML = '### Results ' + '' + 'P(X = ' + k + ') (exactly ' + k + ' successes)' + '' + (pExact * 100).toFixed(4) + '%' + 'P(X ≥ ' + k + ') (at least ' + k + ' successes)' + '' + (pAtLeast * 100).toFixed(4) + '%' + 'P(X ≤ ' + k + ') (at most ' + k + ' successes)' + '' + (pAtMost * 100).toFixed(4) + '%' + 'Expected successes (mean)' + mean.toFixed(4) + '' + 'Standard deviation' + stddev.toFixed(4) + '' + '' + '#### Full Distribution (highlighted row = your k) ' + '' + 'k (successes)P(X = k)' + '' + tableRows + '' + ''; }
#### Formula
This calculator uses the Hypergeometric Distribution:
P(X = k) = C(K, k) × C(N−K, n−k) / C(N, n)
- N = total cards in the deck
- K = total desired cards in the deck
- n = number of cards drawn
- k = desired number of successes in the draw
- C(a, b) = binomial coefficient = a! / (b! × (a−b)!)
Mean: μ = n × (K / N)
Variance: σ² = n × (K/N) × ((N−K)/N) × ((N−n)/(N−1))
Log-space computation is used to handle large factorials accurately.
#### Assumptions & References
- Drawing is done without replacement (hypergeometric model). For drawing with replacement, use the binomial distribution instead.
- Reference: Hypergeometric distribution — Feller, W. (1968). An Introduction to Probability Theory and Its Applications.
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