probs.js

const Decimal = require('decimal.js');

function cdf(n, k, p) {
    const probability = new Decimal(p);
    const lambda = new Decimal(n).times(probability);
    let total = new Decimal(0);
    for (let i = 0; i <= n-k; i++) {
        const res = lambda.pow(i).dividedBy(factorial(i));
        total = total.plus(res);
    }
    return total.times(new Decimal(Math.E).pow(lambda.negated()));
}

function factorial (n) {
    if (n === 0 || n === 1) {
        return 1;
    }
    return n * factorial(n - 1);
}

// Usage: node probs.js <failure pct> <k> <n>
//        node probs.js 10 24 36
const [failurePct, k, n] = process.argv.slice(2);

const failureProbability = new Decimal(failurePct).dividedBy(100);
const kThreshold = new Decimal(k);
const nShards = new Decimal(n);

const cdfValue = cdf(nShards, kThreshold, failureProbability);
const nineCount = ((cdfValue.toString().match(/^0\.(9*)/) || [])[1] || []).length;
const cdfStr = cdfValue.times(100).toString()
console.log(`${failurePct}% / month, k ${kThreshold} / n ${nShards}: ${cdfStr}% (${nineCount} nines)`);