Maximum Number of Dots After Throwing a Dice n Times in C++

Given the task is to calculate the maximum number of dots that can be expected after throwing a dice N times having M faces.

The first face of the dice contains 1 dot, the second face has 2 dots and so on. Likewise the M-th face contains M number of dots.

The probability of appearance of each face becomes 1/M.

Let’s now understand what we have to do using an example −

Input − M=2, N=3

Output − 1.875

Explanation − The dice has 2 sides = {1, 2}

If the dice is thrown 3 times then the sample space will be = M^N = 2³

{(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2,)
(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2,)}
Maximum number in (1, 1, 1) = 1
Maximum number in (1, 1, 2) = 2
Maximum number in (1, 2, 1) = 2
Maximum number in (1, 2, 2) = 2
Maximum number in (2, 1, 1) = 2
Maximum number in (2, 1, 2) = 2
Maximum number in (2, 2, 1) = 2
Maximum number in (2, 2, 2) = 2
Probability of each case = 1/23 = 0.125
Therefore, expectation of maximum number = (1+2+2+2+2+2+2+2) * (0.125) = 1.875

Input − M=2, N=2

Output − 1.75

Approach used in the below program as follows

• The maximum number of cases in which a number can occur can be found using its previous number by using the formula − i^N – (i-1)^N.

  For example if M=4 and N=2, the total number of cases in which maximum = 4 will be 4² – (4-1)² = 7.

  So the final answer will be applying this formula on every element from 1 to M −

  (i * (i^N – (i - 1)^N )) / M^N and adding them up.

• In function MaxExpect() initialize a variable max =0 of type double to store the sum.

• Then loop from i=M till i>0

• Inside the loop apply the above stated formula and keep adding all the resultant values into the variable max.

Example

 Live Demo

#include <bits/stdc++.h>
using namespace std;
double MaxExpect(double M, double N){
   double max = 0.0, i;
   for (i = M; i; i--)
      /*formula to find maximum number and
      sum of maximum numbers*/
      max += (pow(i / M, N) - pow((i - 1) / M, N)) * i;
      return max;
}
int main(){
   double M = 2, N = 3;
   cout << MaxExpect(M, N);
   return 0;
}

Output

If we run the above code we will get the following output −

1.875