Simulate an urn where two balls of the colour drawn are added after each draw; see https://math.stackexchange.com/questions/4636935.

Question4636935.java

package info.joriki.math.stackexchange;

public class Question4636935 {
    final static int ntrials = 10000;
    public static void main(String [] args) {
        int count = 0;
        int total = 0;
        
        for (int n = 0;n < ntrials;n++) {
            int nblack = 1;
            int nred = 1;
            
            boolean firstBlack = false;
            
            for (int i = 0;i < 3;i++) {
                boolean black = Math.random() < nblack / (double) (nblack + nred);
                if (i == 0 && black)
                    firstBlack = true;
                if (black)
                    nblack += 2;
                else
                    nred += 2;
            }
            
            if (nblack <= 5) {
                total++;
                if (firstBlack)
                    count++;
            }
        }
        
        System.out.println(count / (double) total);
    }
}