AP中三个随机选择的数字的概率

给定一个数字n和一个包含1到(2n + 1)个连续数字的数组。随机选择三个元素。查找所选元素位于AP中的概率
例子：

Input : n = 2
Output : 0.4
The array would be {1, 2, 3, 4, 5}
Out of all elements, triplets which 
are in AP: {1, 2, 3}, {2, 3, 4}, 
{3, 4, 5}, {1, 3, 5}
No of ways to choose elements from 
the array: 10 (5C3) 
So, probability = 4/10 = 0.4

Input : n = 5
Output : 0.1515

从(2n + 1)个数字中选择任意3个数字的方式有： ^{(2n +1)} C ₃
现在，要使号码出现在AP中：
共同点为1— {1、2、3}，{2、3、4}，{3、4、5}…{2n-1、2n，2n + 1}
具有共同的区别2 — {1，3，5}，{2，4，6}，{3，5，7}…{2n-3，2n-1，2n + 1}
具有共同的差异n— {1，n + 1，2n + 1}
因此，(2n + 1)个数字中的3个AP组总数为：
(2n – 1)+(2n – 3)+(2n – 5)+…+ 3 +1 = n * n (前n个奇数的总和为n * n)
因此，在(2n + 1)个连续数字中3个随机选择的数字出现在AP中的概率=(n * n)/ ^{(2n + 1)} C ₃ = 3 n /(4(n * n)– 1)

C++

// CPP program to find probability that
// 3 randomly chosen numbers form AP.
#include 
using namespace std;
 
// function to calculate probability
double procal(int n)
{
    return (3.0 * n) / (4.0 * (n * n) - 1);
}
 
// Driver code to run above function
int main()
{
    int a[] = { 1, 2, 3, 4, 5 };
    int n = sizeof(a)/sizeof(a[0]);
    cout << procal(n);
    return 0;
}

Java

// Java program to find probability that
// 3 randomly chosen numbers form AP.
 
class GFG {
     
    // function to calculate probability
    static double procal(int n)
    {
        return (3.0 * n) / (4.0 * (n * n) - 1);
    }
 
    // Driver code to run above function
    public static void main(String arg[])
    {
        int a[] = { 1, 2, 3, 4, 5 };
        int n = a.length;
        System.out.print(Math.round(procal(n) * 1000000.0) / 1000000.0);
    }
}
 
// This code is contributed by Anant Agarwal.

Python3

# Python3 program to find probability that
# 3 randomly chosen numbers form AP.
 
# Function to calculate probability
def procal(n):
 
    return (3.0 * n) / (4.0 * (n * n) - 1)
 
# Driver code
a = [1, 2, 3, 4, 5]
n = len(a)
print(round(procal(n), 6))
 
# This code is contributed by Smitha Dinesh Semwal.

C#

// C# program to find probability that
// 3 randomly chosen numbers form AP.
using System;
 
class GFG {
     
    // function to calculate probability
    static double procal(int n)
    {
        return (3.0 * n) / (4.0 * (n * n) - 1);
    }
 
    // Driver code
    public static void Main()
    {
        int []a = { 1, 2, 3, 4, 5 };
        int n = a.Length;
        Console.Write(Math.Round(procal(n) *
                    1000000.0) / 1000000.0);
    }
}
 
// This code is contributed by nitin mittal

Javascript

// Javascript program to find probability that
// 3 randomly chosen numbers form AP.
 
// function to calculate probability
function procal(n)
{
    return (3.0 * n) /
           (4.0 * (n *
              n) - 1);
}
 
    // Driver code
    let a = [1, 2, 3, 4, 5];
    let n = a.length;
    document.write(procal(n));
     
// This code is contributed by _saurabh_jaiswal

输出：

0.151515

时间复杂度：O(1)