Probability of getting two consecutive heads after choosing a random coin among two different types of coins

Given two coins which have probability of getting heads p% and q% respectively, the task is to determine the probability of getting two consecutive heads after choosing random coins among the given coins.

Examples:

Input: p = 33, q = 66
Output: 0.550000000000000



Input: p = 33, q = 66
Output: 0.550000000000000

Approach:
Since both the coins are not identical so Bayes’s theorem will be used to get the desired probability.
As the coins are to be chosen randomly then any of them can be chosen so p and q will be included in the calculation. After applying Bayes’s theorem, the required answer will be (p * p + q * q) / (p + q) because if the first coin is chosen then the probability of getting both heads back to back is p * p and same for the second coin.
It’s an application of Bayes’ theorem.

P(B | A) = P(A |^| B) / P(A) = (1/2 * p * p + 1/2 * q * q) / (1/2 * p + 1/2 * q) = (p * p + q * q) / (p + q) where;
P(B) = probability to get heads on the second throw,
P(A) = probability to get heads on the first throw and
P(A |^| B) = probability to get heads on both the throws.
So, P(B | A) is the probability to get heads on the second throw if we are given that we got heads
on the first one.
Here A, B denotes 1st and 2nd coin.

Below is the implementation of above approach:

C++

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// C++ program to get the probability
// of getting two consecutive heads
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the probability
// of getting two consecutive heads
double getProbability(double p, double q)
{
    p /= 100;
    q /= 100;
  
    // Formula derived from Bayes's theorem
    double probability = (p * p + q * q) / (p + q);
    return probability;
}
  
// Driver code
int main()
{
    double p, q;
  
    // given the probability of getting
    // a head for both the coins
    p = 80;
    q = 40;
  
    cout << fixed
         << setprecision(15)
         << getProbability(p, q)
         << endl;
  
    return 0;
}

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Java

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// Java program to get the probability
// of getting two consecutive heads
  
import java.io.*;
  
class GFG {
    
  
// Function to return the probability
// of getting two consecutive heads
static double getProbability(double p, double q)
{
    p /= 100;
    q /= 100;
  
    // Formula derived from Bayes's theorem
    double probability = (p * p + q * q) / (p + q);
    return probability;
}
  
// Driver code
  
  
    public static void main (String[] args) {
            double p, q;
  
    // given the probability of getting
    // a head for both the coins
    p = 80;
    q = 40;
  
     System.out.println( getProbability(p, q));
    }
}
// This code is contributed by  anuj_67..

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Python 3

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# Python 3 program to get the probability
# of getting two consecutive heads
  
# Function to return the probability
# of getting two consecutive heads
def getProbability(p, q):
  
    p /= 100
    q /= 100
  
    # Formula derived from Bayes's theorem
    probability = (p * p + q * q) / (p + q)
    return probability
  
# Driver code
if __name__ == "__main__":
  
    # given the probability of getting
    # a head for both the coins
    p = 80
    q = 40
  
    print(getProbability(p, q))
  
# This code is contributed 
# by ChitraNayal

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C#

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// C# program to get the probability
// of getting two consecutive heads
using System;
  
class GFG {
  
  
// Function to return the probability
// of getting two consecutive heads
static double getProbability(double p, double q)
{
    p /= 100;
    q /= 100;
  
    // Formula derived from Bayes's theorem
    double probability = (p * p + q * q) / (p + q);
    return probability;
}
  
// Driver code
  
  
    public static void Main () {
            double p, q;
  
    // given the probability of getting
    // a head for both the coins
    p = 80;
    q = 40;
  
    Console.WriteLine( getProbability(p, q));
    }
}
// This code is contributed by inder_verma..

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PHP

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<?php
// PHP program to get the probability
// of getting two consecutive heads
  
// Function to return the probability
// of getting two consecutive heads
function getProbability($p, $q)
{
    $p /= 100;
    $q /= 100;
  
    // Formula derived from 
    // Bayes's theorem
    $probability = ($p * $p + $q * $q) / 
                             ($p + $q);
    return $probability;
}
  
// Driver code
  
// given the probability of getting
// a head for both the coins
$p = 80;
$q = 40;
  
echo getProbability($p, $q);
  
// This code is contributed
// by Shivi_Aggarwal
?>

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Output:

0.666666666666667


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