SameBirthday

MonteCarlo.cs

using System;

namespace MonteCarlo
{
    class Program
    {
        private static Random r = new Random();
        private static readonly double GestationPeriodMean = 280.6;
        private static readonly double GestationPeriodStandardDeviation = 9.7;
        private static readonly int Simulations = 10000000;
        

        static void Main(string[] args)
        {
            double conceptionDifferenceDelta = 0;
            bool sharedBirthday = true;
            while (sharedBirthday)
            {
                var sameDayBirths = GetSameDayBirthCount(conceptionDifferenceDelta);
                Console.WriteLine(string.Format("ConceptionDateDiff:{0} {1:P2}", conceptionDifferenceDelta, (double)sameDayBirths / Simulations));

                sharedBirthday = sameDayBirths > 0;
                conceptionDifferenceDelta += 1;
            }
        }

        private static int GetSameDayBirthCount(double conceptionDifferenceDelta)
        {
            int sameDayBirths = 0;
            for (int i = 0; i < Simulations; i++)
            {
                double birthDateFirst = getBirthDate();
                double birthDateSecond = getBirthDate();

                // move the conception date of second by conceptionDifferenceDelta days
                birthDateSecond += conceptionDifferenceDelta;

                if ((int) birthDateFirst == (int) birthDateSecond)
                    sameDayBirths++;
            }
            return sameDayBirths;
        }


        private static double getBirthDate()
        {
            // Box-Muller transform - http://stackoverflow.com/a/218600/3006667
            double u1 = 1.0 - r.NextDouble(); //uniform(0,1] random doubles
            double u2 = 1.0 - r.NextDouble();
            double randStdNormal = Math.Sqrt(-2.0 * Math.Log(u1)) *
                         Math.Sin(2.0 * Math.PI * u2); //random normal(0,1)
            double result = GestationPeriodMean + GestationPeriodStandardDeviation * randStdNormal; //random normal(mean,stdDev^2)
            return result;
        }
        
    }
}

Output

ConceptionDateDiff:0 2.91 %
ConceptionDateDiff:1 2.90 %
ConceptionDateDiff:2 2.88 %
ConceptionDateDiff:3 2.84 %
ConceptionDateDiff:4 2.79 %
ConceptionDateDiff:5 2.73 %
ConceptionDateDiff:6 2.64 %
ConceptionDateDiff:7 2.55 %
ConceptionDateDiff:8 2.44 %
ConceptionDateDiff:9 2.35 %
ConceptionDateDiff:10 2.24 %
ConceptionDateDiff:11 2.11 %
ConceptionDateDiff:12 1.99 %
ConceptionDateDiff:13 1.85 %
ConceptionDateDiff:14 1.73 %
ConceptionDateDiff:15 1.59 %
ConceptionDateDiff:16 1.47 %
ConceptionDateDiff:17 1.35 %
ConceptionDateDiff:18 1.23 %
ConceptionDateDiff:19 1.12 %
ConceptionDateDiff:20 1.01 %
ConceptionDateDiff:21 0.90 %
ConceptionDateDiff:22 0.80 %
ConceptionDateDiff:23 0.72 %
ConceptionDateDiff:24 0.63 %
ConceptionDateDiff:25 0.55 %
ConceptionDateDiff:26 0.48 %
ConceptionDateDiff:27 0.42 %
ConceptionDateDiff:28 0.36 %
ConceptionDateDiff:29 0.31 %
ConceptionDateDiff:30 0.27 %

The length of human pregnancy as calculated by ultrasonographic measurement of the fetal biparietal diameter (H. Kieler, O. Axelsson, S. Nilsson, U. Waldenströ) has a very nice table of 10 published studies, which found various means between 272-283 days. The authors contribute their own results as well, recording over 800 mothers who went into labor spontaneously. When pregnancy length was calculated using ultrasound in the second trimester the mean was 280.6 days, standard deviation 9.7 days. When it was calculated using LMP, the mean was 283.6 days with a standard deviation of 10.5 days.