I was looking at a question that was talking about a bad implementation of the Fisher-Yates shuffling algorithm and I was perplexed that there was a bias when implemented incorrectly.

The two algorithms are these:

private Random _random = new Random();

public int[] FisherYates(int[] source)
{
    int[] output = source.ToArray();
    for (var i = 0; i < output.Length; i++)
    {
        var j = _random.Next(i, output.Length);
        (output[i], output[j]) = (output[j], output[i]);
    }
    return output;
}

public int[] FisherYatesBad(int[] source)
{
    int[] output = source.ToArray();
    for (var i = 0; i < output.Length; i++)
    {
        var j = _random.Next(0, output.Length);
        (output[i], output[j]) = (output[j], output[i]);
    }
    return output;
}

A really subtle different, but enough to cause a massive bias.

Good implementation:

Bad implementation:

To be clear about these plots, I start with the numbers 0 to 99, create 10_000_000 shuffles using whichever algorithm and then I average the values in each of the shuffles to get a single set of figures. If the shuffle is trying random then all 100 figures would belong to the same normal distribution.

Now, that's all fine, but I thought I'd check to see if these methods produce valid results:

public int[] OrderByRandomNext(int[] source) => source.OrderBy(x => _random.Next()).ToArray();

public int[] OrderByRandomNextDouble(int[] source) => source.OrderBy(x => _random.NextDouble()).ToArray();

Both nice a neat, but are they a fair shuffle?

OrderByRandomNext:

OrderByRandomNextDouble:

Notice that the 1 and the 100 figures are significantly lower in each?

Well, I thought it might be an artefact of how OrderBy works. So I tested it with another random number generator - one brought to use from Eric Lippert in his improving Random series.

public int[] OrderByBetterRandomNextDouble(int[] source) => source.OrderBy(x => BetterRandom.NextDouble()).ToArray();

public static class BetterRandom
{
    private static readonly ThreadLocal<RandomNumberGenerator> crng =
        new ThreadLocal<RandomNumberGenerator>(RandomNumberGenerator.Create);

    private static readonly ThreadLocal<byte[]> bytes =
        new ThreadLocal<byte[]>(() => new byte[sizeof(int)]);

    public static int NextInt()
    {
        crng.Value.GetBytes(bytes.Value);
        return BitConverter.ToInt32(bytes.Value, 0) & int.MaxValue;
    }

    public static double NextDouble()
    {
        while (true)
        {
            long x = NextInt() & 0x001FFFFF;
            x <<= 31;
            x |= (long)NextInt();
            double n = x;
            const double d = 1L << 52;
            double q = n / d;
            if (q != 1.0)
                return q;
        }
    }
}

Well, here's the chart:

There's no bias!

Here's my code to generate the data (run in LINQPad):

void Main()
{
    var n = 100;
    var s = 1000000;

    var numbers = Enumerable.Range(0, n).ToArray();

    var algorithms = new Func<int[], int[]>[]
    {
        FisherYates,
        OrderByRandomNext,
        OrderByRandomNextDouble,
        OrderByBetterRandomNextDouble,
    };

    var averages =
        algorithms
            .Select(algorithm =>
                Enumerable
                    .Range(0, numbers.Length)
                    .Select(x =>
                        Enumerable
                            .Range(0, s)
                            .Select(y => algorithm(numbers))
                            .Aggregate(0.0, (a, v) => a + (double)v[x] / s))
                    .ToArray())
            .Select(x => new
            {
                averages = x,
                distribution = Accord.Statistics.Distributions.Univariate.NormalDistribution.Estimate(x.Skip(1).SkipLast(1).ToArray()),
                first = x.First(),
                last = x.Last(),
            })
            .Select(x => new
            {
                x.averages,
                x.distribution,
                x.first,
                x.last,
                first_prob =x.distribution.DistributionFunction(x.first),
                last_prob = x.distribution.DistributionFunction(x.last),
            })
            .ToArray();

    var d = 

    averages.Dump();
}

private Random _random = new Random();

    public int[] FisherYates(int[] source)
    {
        int[] output = source.ToArray();
        for (var i = 0; i < output.Length; i++)
        {
            var j = _random.Next(i, output.Length);
            (output[i], output[j]) = (output[j], output[i]);
        }
        return output;
    }

public int[] OrderByRandomNext(int[] source) => source.OrderBy(x => _random.Next()).ToArray();

public int[] OrderByRandomNextDouble(int[] source) => source.OrderBy(x => _random.NextDouble()).ToArray();

    public int[] OrderByBetterRandomNextDouble(int[] source) => source.OrderBy(x => BetterRandom.NextDouble()).ToArray();

    public static class BetterRandom
    {
        private static readonly ThreadLocal<RandomNumberGenerator> crng =
            new ThreadLocal<RandomNumberGenerator>(RandomNumberGenerator.Create);

        private static readonly ThreadLocal<byte[]> bytes =
            new ThreadLocal<byte[]>(() => new byte[sizeof(int)]);

        public static int NextInt()
        {
            crng.Value.GetBytes(bytes.Value);
            return BitConverter.ToInt32(bytes.Value, 0) & int.MaxValue;
        }

        public static double NextDouble()
        {
            while (true)
            {
                long x = NextInt() & 0x001FFFFF;
                x <<= 31;
                x |= (long)NextInt();
                double n = x;
                const double d = 1L << 52;
                double q = n / d;
                if (q != 1.0)
                    return q;
            }
        }
    }

Here's the data that I generated:

distribution                                             | first              | last               | first_prob             | last_prob            
-------------------------------------------------------- | ------------------ | ------------------ | ---------------------- | ---------------------
N(x; μ = 49.50267467345823, σ² = 0.0008896228453062147)  | 49.505465999987585 | 49.49833699998965  | 0.5372807100387846     | 0.44218570467529394  
N(x; μ = 49.50503062243786, σ² = 0.0009954477334487531)  | 49.36330799998817  | 49.37124399998651  | 3.529550818615057E-06  | 1.115772521409486E-05
N(x; μ = 49.505720877539765, σ² = 0.0008257970106087029) | 49.37231699998847  | 49.386660999990106 | 1.7228855271333998E-06 | 1.712972513601141E-05
N(x; μ = 49.49994663264188, σ² = 0.0007518765247716318)  | 49.50191999998847  | 49.474235999989205 | 0.5286859991636343     | 0.17421285127499514  

Here's my question. What's up with System.Random and the bias it introduces?