Based on https://www.desmos.com/calculator/ycoftrodqx

GetPassFailRateWithConfidence.cs

void Main()
{
    GetPassFailRateWithConfidence(10, 2, 0.95).ToString().Dump();
    GetPassFailRateWithConfidence(100, 20, 0.95).ToString().Dump();
    GetPassFailRateWithConfidence(1000, 200, 0.95).ToString().Dump();
    GetPassFailRateWithConfidence(10000, 2000, 0.95).ToString().Dump();
}

public ErrorRate GetPassFailRateWithConfidence(long passes, long fails, double confidence)
{
    double fact(long n) => Math.Round(Math.Sqrt(Math.PI * (2 * n + 1.0 / 3)) * Math.Pow(n / Math.E, n));

    double nCr(long n, long r) =>
        fact(n).Dump() / (fact(r) * fact(n - r));

    long gcd(long a, long b) => b > 0
        ? gcd(b, a % b)
        : a;

    double nCr2(long n, long r, double factor)
    {
        if (r > n / 2) r = n - r;

        double ans = factor;
        for (long i = 1; i <= r; i++)
        {
            var num = (n - r + i);
            var divisor = gcd(num, i);
            ans *= (num / divisor);
            ans /= (i / divisor);
        }

        return ans;
    }

    var total = passes + fails;
    var θ = (passes + 0.5) / (total + 1);

    double P(long p) =>
        nCr2(total, p, Math.Pow(θ, p) * Math.Pow(1 - θ, total - p));

    var minThreshold = (1 - confidence) / 2;
    var maxThreshold = 1 - minThreshold;
    var sum = 0.0;
    double? min = default;
    double? max = default;
    for (var i = 0; i <= total; i++)
    {
        if (sum >= maxThreshold && max == null)
        {
            max = (double)i / total;
            break;
        }

        sum += P(i);
        //sum.Dump();

        if (sum >= minThreshold && min == null)
        {
            min = (double)i / total;
        }
    }

    if (max == null)
    {
        max = 1;   
    }

    return new ErrorRate(mean: (double)passes / total, min: min.Value, max: max.Value, confidence: confidence);
}

public class ErrorRate
{
    public ErrorRate(double mean, double min, double max, double confidence)
    {
        this.Mean = mean;
        this.Min = min;
        this.Max = max;
        this.Confidence = confidence;
    }

    public double Mean { get; }
    public double Min { get; }
    public double Max { get; }
    public double Confidence { get; }

    public override string ToString()
    {
        return $"{this.Mean:P} ({this.Min:P}-{this.Max:P} @{this.Confidence:P})";
    }
}