Probabilidad de obtener dos caras consecutivas al elegir una moneda al azar entre dos tipos de monedas diferentes

Dadas dos monedas que tienen probabilidad de obtener cara p% y q% respectivamente, la tarea es determinar la probabilidad de obtener dos caras consecutivas después de elegir monedas al azar entre las monedas dadas.
Ejemplos: 
 

Entrada: p = 33, q = 66 
Salida: 0,550000000000000
Entrada: p = 33, q = 66 
Salida: 0,5500000000000000 
 

Enfoque: 
dado que ambas monedas no son idénticas , se utilizará  el teorema de Bayes para obtener la probabilidad deseada.
Como las monedas se elegirán al azar, se puede elegir cualquiera de ellas, por lo que p y q se incluirán en el cálculo. Después de aplicar el teorema de Bayes, la respuesta requerida será (p * p + q * q) / (p + q) porque si se elige la primera moneda, entonces la probabilidad de obtener ambas caras espalda con espalda es p * p y lo mismo para la segunda moneda 
Es una aplicación del teorema de Bayes. 
 

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) donde; 
P(B) = probabilidad de obtener cara en el segundo lanzamiento, 
P(A) = probabilidad de obtener cara en el primer lanzamiento y 
P(A |^| B) = probabilidad de obtener cara en ambos lanzamientos. 
Entonces, P(B | A) es la probabilidad de sacar cara en el segundo tiro si sabemos que salimos cara 
en el primero. 
Aquí A, B denota la primera y la segunda moneda. 
 

A continuación se muestra la implementación del enfoque anterior: 
 

// 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;
}

// 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..

# 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

// 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..

<?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
?>

<script>
 
// Javascript 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
    let probability = (p * p + q * q) / (p + q);
    return probability;
}
 
// Driver code
 
 
// given the probability of getting
// a head for both the coins
let p = 80.0;
let q = 40.0;
 
document.write(getProbability(p, q).toPrecision(15));
 
 
</script>

0.666666666666667

Complejidad de tiempo: O(1)

Espacio Auxiliar: O(1)