Simulate the difference between the expected maximum length of a run of heads and the expected maximum length of a run of heads or tails; see https://math.stackexchange.com/questions/3464662.

Question3464662.java

import java.util.Random;

public class Question3464662 {
    final static long ntrials = 10000000;
    final static Random random = new Random ();
    
    public static void main(String [] args) {
        long headCounts = 0;
        long bothCounts = 0;
        for (long n = 0;n < ntrials;n++) {
            int headCount = 0;
            int tailCount = 0;
            int maxHeadCount = 0;
            int maxTailCount = 0;
            for (int i = 0;i < 1000;i++) {
                boolean heads = random.nextBoolean();
                if (heads) {
                    headCount++;
                    tailCount = 0;
                    maxHeadCount = Math.max(maxHeadCount, headCount);
                }
                else {
                    tailCount++;
                    headCount = 0;
                    maxTailCount = Math.max(maxTailCount, tailCount);
                }
            }
            headCounts += maxHeadCount;
            bothCounts += Math.max(maxHeadCount, maxTailCount);
        }
        
        System.out.println(headCounts / (double) ntrials);
        System.out.println(bothCounts / (double) ntrials);
        System.out.println((bothCounts - headCounts) / (double) ntrials);
    }
}