Probability for three randomly chosen numbers to be in AP (original) (raw)

Last Updated : 21 Sep, 2022

Given a number n and an array containing 1 to (2n+1) consecutive numbers. Three elements are chosen at random. Find the probability that the elements chosen are in A.P.

Examples:

Input : n = 2
Output : 0.4
Explanation:
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

The number of ways to select any 3 numbers from (2n+1) numbers are:(2n + 1) C 3
Now, for the numbers to be in AP:
with common difference 1---{1, 2, 3}, {2, 3, 4}, {3, 4, 5}...{2n-1, 2n, 2n+1}
with common difference 2---{1, 3, 5}, {2, 4, 6}, {3, 5, 7}...{2n-3, 2n-1, 2n+1}
with common difference n--- {1, n+1, 2n+1}
Therefore, Total number of AP group of 3 numbers in (2n+1) numbers are:
(2n - 1)+(2n - 3)+(2n - 5) +...+ 3 + 1 = n * n (Sum of first n odd numbers is n * n )
So, probability for 3 randomly chosen numbers in (2n + 1) consecutive numbers to be in AP = (n * n) / (2n + 1) C 3 = 3 n / (4 (n * n) - 1)

Below is the implementation of the above approach:

C++ `

// CPP program to find probability that // 3 randomly chosen numbers form AP. #include <bits/stdc++.h> 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

PHP

JavaScript

`

Time Complexity: O(1)
Auxiliary Space: O(1)