Relative prime numbers

Write a program, RelativelyPrime_YI.java to prompt the user for an integer, n. The program displays an n-by-n table such a that there is an * in row i and column j if the gcd of i and j is 1 ( i and j are relatively prime) and a space in that position otherwise.

Hint: use GCD_YI.java to finds the greatest common divisor of two integers

Example output:

>run ca_relativelyprime 2 20
   1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 
 1 * * * * * * * * * * * * * * * * * * * * 
 2 *   *   *   *   *   *   *   *   *   *   
 3 * *   * *   * *   * *   * *   * *   * * 
 4 *   *   *   *   *   *   *   *   *   *   
 5 * * * *   * * * *   * * * *   * * * *   
 6 *       *   *       *   *       *   *   
 7 * * * * * *   * * * * * *   * * * * * * 
 8 *   *   *   *   *   *   *   *   *   *   
 9 * *   * *   * *   * *   * *   * *   * * 
10 *   *       *   *   *   *       *   *   
11 * * * * * * * * * *   * * * * * * * * * 
12 *       *   *       *   *       *   *   
13 * * * * * * * * * * * *   * * * * * * * 
14 *   *   *       *   *   *   *   *   *   
15 * *   *     * *     *   * *   * *   *   
16 *   *   *   *   *   *   *   *   *   *   
17 * * * * * * * * * * * * * * * *   * * * 
18 *       *   *       *   *       *   *   
19 * * * * * * * * * * * * * * * * * *   * 
20 *   *       *   *   *   *       *   *   
> 
*/

Modify Gambler.java to take an extra command line parameter that specifies the number of bets the gambler is willing to make so that there are three possible ways for the game to end:
* the gambler wins, loses, or runs “out of time” meaning the gambler gives up.
* Add to the output to give the expected amount of money the gambler will have when the game ends.

Think of the different outcomes and how would report on them. That means what your output would look like and how you would display it. Look back to the format of the previous gambler’s ruin activity.

A random(get it?) example:

stake: $5 goal: $10 trials: 10 games and run out of time is 7 bets

1 game – lost all your money
2 game – won
3 game – end by the 7th bet and you had $6
4 game – lost all your money
5 game – end by the 7th bet and you had $4
6 game – lost all your money
7 game – won
8 game – end by the 7th bet and you had $2
9 game – won
10 game – lost all your money

6 + 4 + 2 = 12 total accumulated from “run out of time”
the expected amount of money from 10 games is 12/3 = $4

Lesson slides on page 28

public class Gambler
{
    public static void main(String[] args)
   {
     int stake = Integer.parseInt(args[0]);
     int goal = Integer.parseInt(args[1]);
     int trials = Integer.parseInt(args[2]);

    int wins = 0;
    for (int i = 0; i < trials; i++)
     {
       int t = stake;
       while (t > 0 && t < goal)
        {
            if (Math.random() < 0.5) t++;
            else t--;
        }
        if (t == goal) wins++;
     }
   StdOut.println(wins + " wins of " + trials);  
   }
}

WARNING: your program should run only once and produce this chart:
a. Modify the original gambler to show how the increase of “trials” affects the probability of winning.

A good easy format:
stake = 5 goal = 25

Trials Wins Probability
100 15%
1000 19%
10000 22%
100000 20%
1000000 20%

WARNING: your program should run only once and produce this chart:
b. Modify the original gambler to show how the increase of the “stake” affects the probability of winning.

A good easy format:
goal = 25 trials = 1000

Stake Wins Probability
5 17%
10 39%
15
20

WARNING: your program should run only once and produce this chart:
c. Modify the original gambler to show how the increase of the “goal” affects the probability of winning.

A good easy format:
stake = 5 trials = 1000

Goal Wins Probability
5 98%
10 50%
15
20
25

From page 28 on this pdf:
1. What is the “if (t == goal) wins++; ” keeping track of?
2. What does the outer loop control?
3. What happens at each trial?
4. What is the identifier “win” for?
5. What is the “stake” for?
6. What is the “goal” for?
7. How many trials would produce a good result?
8. How do you read the output in plain English?
% java Gambler 5 25 1000
203 wins of 1000

1. If you were running this program to find the probability of winning in this game, what would need to be done to come up with the best approximation to the actual value?

a. Implement another loop that will increase the goal by doubling it at every iteration and report the conclusion. Keep all other variables fixed. The stake should be $5.

b. Implement another loop that will increase the trials by factors of 10 it at every iteration and report the conclusion. Keep all other variables fixed. The stake should be $5.

Challenge (Optional):

Add what is needed to be able to systematically increase both the goals and the trials. Report your result in full a sentence.

public class Gambler
{
    public static void main(String[] args)
   {
     int stake = Integer.parseInt(args[0]);
     int goal = Integer.parseInt(args[1]);
     int trials = Integer.parseInt(args[2]);

    int wins = 0;
    for (int i = 0; i < trials; i++)
     {
       int t = stake;
       while (t > 0 && t < goal)
        {
            if (Math.random() < 0.5) t++;
            else t--;
        }
        if (t == goal) wins++;
     }
   StdOut.println(wins + " wins of " + trials);  
   }
}

Write a program Checkerboard_YI.java that takes one command-line argument N and prints out a two dimensional N-by-N checkerboard pattern with alternating spaces and asterisks, like the following 4-by-4 pattern.