彩票中奖机会练习

Question

So I have a problem that I'm stuck on it since 3 days ago.

You want to participate at the lottery 6/49 with only one winning variant(simple) and you want to know what odds of winning you have:

-at category I (6 numbers)

-at category II (5 numbers)

-at category III (4 numbers)

Write a console app which gets from input the number of total balls, the number of extracted balls, and the category, then print the odds of winning with a precision of 10 decimals if you play with one simple variant.

Inputs:

40

5

II

Result I must print:

0.0002659542

static void Main(string[] args)
        {
            int numberOfBalls = Convert.ToInt32(Console.ReadLine());
            int balls = Convert.ToInt32(Console.ReadLine());
            string line = Console.ReadLine();
            int theCategory = FindCategory(line);
            double theResult = CalculateChance(numberOfBalls, balls, theCategory);
            Console.WriteLine(theResult);
        }
        static int FindCategory (string input)
        {
            int category = 0;
            switch (input)
            {
                case "I":
                    category = 1;
                    break;
                case "II":
                    category = 2;
                    break;
                case "III":
                    category = 3;
                    break;
                default:
                    Console.WriteLine("Wrong category.");
                    break;
            }
            return category;
        }
        static int CalculateFactorial(int x)
        {
            int factorial = 1;
            for (int i = 1; i <= x; i++)
                factorial *= i;
            return factorial;
        }
        static int CalculateCombinations(int x, int y)
        {
            int combinations = CalculateFactorial(x) / (CalculateFactorial(y) * CalculateFactorial(x - y));
            return combinations;
        }
        static double CalculateChance(int a, int b, int c)
        {
            double result = c / CalculateCombinations(a, b);
            return result;
        }

Now my problems: I'm pretty sure I have to use Combinations. For using combinations I need to use Factorials. But on the combinations formula I'm working with pretty big factorials so my numbers get truncated. And my second problem is that I don't really understand what I have to do with those categories, and I'm pretty sure I'm doing wrong on that method also. I'm new to programming so please bare with me. And I can use for this problem just basic stuff, like conditions, methods, primitives, arrays.

Answer 1

Let's start from combinatorics; first, come to terms:

• a - all possible numbers ( 40 in your test case)
• t - all taken numbers ( 5 in your test case)
• c - category ( 2 ) in your test case

So we have

t - c + 1 for numbers which win and c - 1 for numbers which lose. Let's count combinations:

All combinations: take t from a possible ones:

A = a! / t! / (a - t)! 

Winning numbers' combinations: take t - c + 1 winning number from t possible ones:

W = t! / (t - c + 1)! / (t - t + c - 1) = t! / (t - c + 1)! / (c - 1)!

Lost numbers' combinations: take c - 1 losing numbers from a - t possible ones:

L = (a - t)! / (c - 1)! / (a - t - c + 1)!

All combinations with category c , ie with exactly t - c + 1 winning and c - 1 losing numbers:

C = L * W

Probability:

P = C / A = L * W / A =

t! * t! (a - t)! * (a - t)! / (t - c + 1)! / (c - 1)! / (c - 1)! / (a - t- c + 1)! / a!

Ugh: Not let's implement some code for it:

Code:

// double : note, that int is too small for 40! and the like values
private static double Factorial(int value) {
  double result = 1.0;

  for (int i = 2; i <= value; ++i)
    result *= i;

  return result;
}

private static double Chances(int a, int t, int c) =>
  Factorial(a - t) * Factorial(t) * Factorial(a - t) * Factorial(t) /
    Factorial(t - c + 1) /
    Factorial(c - 1) / 
    Factorial(c - 1) /
    Factorial(a - t - c + 1) /
    Factorial(a); 

Test:

 Console.Write(Chances(40, 5, 2));

Outcome:

 0.00026595421332263435

Edit:

in terms of Combinations , if C(x, y) which means "take y items from x " we have

A = C(a, t); W = C(t, t - c + 1); L = C(a - t, c - 1)

and

P = W * L / A = C(t, t - c + 1) * C(a - t, c - 1) / C(a, t)

Code for Combinations is quite easy; the only trick is that we return double :

// Let'g get rid of noisy "Compute": what else can we do but compute?
// Just "Combinations" without pesky prefix.
static double Combinations(int x, int y) =>
  Factorial(x) / Factorial(y) / Factorial(x - y);

private static double Chances(int a, int t, int c) =>
  Combinations(t, t - c + 1) *
  Combinations(a - t, c - 1) /
  Combinations(a, t); 

You can fiddle the solution