Find probability that a player wins when probabilities of hitting the target are given

0 0 4 minutes read

Given four integers p, q, r, and s. Two players are playing a game where both the players hit a target and the first player who hits the target wins the game. The probability of the first player hitting the target is p / q and that of the second player hitting the target is r / s. The task is to find the probability of the first player winning the game.

Examples:

Input: p = 1, q = 4, r = 3, s = 4
Output: 0.307692308

Input: p = 1, q = 2, r = 1, s = 2
Output: 0.666666667

Approach: The probability of the first player hitting the target is p / q and missing the target is 1 – p / q.
The probability of the second player hitting the target is r / s and missing the target is 1 – r / s.
Let the first player be x and the second player is y.
So the total probability will be x won + (x lost * y lost * x won) + (x lost * y lost * x lost * y lost * x won) + … so on.
Because x can win at any turn, it’s an infinite sequence.
Let t = (1 – p / q) * (1 – r / s). Here t < 1 as p / q and r / s are always <1.
So the series will become, p / q + (p / q) * t + (p / q) * t² + …
This is an infinite GP series with a common ratio of less than 1 and its sum will be (p / q) / (1 – t).

Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the probability of the winner
double find_probability(double p, double q,
                        double r, double s)
{
 
    double t = (1 - p / q) * (1 - r / s);
 
    double ans = (p / q) / (1 - t);
 
    return ans;
}
 
// Driver Code
int main()
{
    double p = 1, q = 2, r = 1, s = 2;
 
    // Will print 9 digits after the decimal point
    cout << fixed << setprecision(9)
         << find_probability(p, q, r, s);
 
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
import java.text.DecimalFormat;
 
class solution
{
 
// Function to return the probability of the winner
static double find_probability(double p, double q,
                        double r, double s)
{
 
    double t = (1 - p / q) * (1 - r / s);
 
    double ans = (p / q) / (1 - t);
 
    return ans;
}
 
// Driver Code
public static void main(String args[])
{
    double p = 1, q = 2, r = 1, s = 2;
 
    // Will print 9 digits after the decimal point
    DecimalFormat dec = new DecimalFormat("#0.000000000");
    System.out.println(dec.format(find_probability(p, q, r, s)));
}
}
// This code is contributed by
// Surendra_Gangwar

Python3

# Python3 implementation of the approach 
 
# Function to return the probability
# of the winner 
def find_probability(p, q, r, s) :
     
    t = (1 - p / q) * (1 - r / s)
 
    ans = (p / q) / (1 - t); 
 
    return round(ans, 9)
 
# Driver Code 
if __name__ == "__main__" : 
 
    p, q, r, s = 1, 2, 1, 2
 
    # Will print 9 digits after 
    # the decimal point 
    print(find_probability(p, q, r, s)) 
 
# This code is contributed by Ryuga

C#

// C# implementation of the approach
using System;
 
class GFG
{
 
// Function to return the probability of the winner
static double find_probability(double p, double q,
                        double r, double s)
{
 
    double t = (1 - p / q) * (1 - r / s);
 
    double ans = (p / q) / (1 - t);
 
    return ans;
}
 
// Driver Code
public static void Main()
{
    double p = 1, q = 2, r = 1, s = 2;
    Console.WriteLine(find_probability(p, q, r, s));
}
}
 
// This code is contributed by
// anuj_67..

PHP

<?php
// PHP implementation of the approach
 
// Function to return the probability
// of the winner
function find_probability($p, $q, $r, $s)
{
    $t = (1 - $p / $q) * (1 - $r / $s);
 
    $ans = ($p / $q) / (1 - $t);
 
    return $ans;
}
 
// Driver Code
$p = 1; $q = 2;
$r = 1; $s = 2;
 
// Will print 9 digits after 
// the decimal point
$res = find_probability($p, $q, $r, $s);
$update = number_format($res, 7);
echo $update;
 
// This code is contributed by Rajput-Ji
?>

Javascript

<script>
 
 
// Javascript implementation of the approach
 
// Function to return the probability of the winner
function find_probability(p, q, r, s)
{
 
    var t = (1 - p / q) * (1 - r / s);
 
    var ans = (p / q) / (1 - t);
 
    return ans;
}
 
// Driver Code
var p = 1, q = 2, r = 1, s = 2;
// Will print 9 digits after the decimal point
document.write( find_probability(p, q, r, s).toFixed(9));
 
 
</script>

Output:

0.666666667

Time Complexity: O(1)

Auxiliary Space: O(1)

Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 zambiatek!