Given a rectangular matrix, we can move from current cell in 4 directions with equal probability. The 4 directions are right, left, top or bottom. Calculate the Probability that after N moves from a given position (i, j) in the matrix, we will not cross boundaries of the matrix at any point.
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The idea is to perform something similar to DFS. We recursively traverse in each of the 4 allowed direction and for each cell encountered, we calculate the required probability with one less move. As each direction has equal probability, each direction will contribute to 1/4 of total probability of that cell i.e. 0.25. We return 0 if we step outside the matrix and return 1 if N steps are completed without crossing matrix boundaries.
Below is the implementation of above idea :
C++
/// C++ program to find the probability // that we do not cross boundary of a // matrix after N moves. #include <iostream> using namespace std;
// check if (x, y) is valid matrix coordinate bool isSafe(int x, int y, int m, int n)
{ return (x >= 0 && x < m &&
y >= 0 && y < n);
} // Function to calculate probability // that after N moves from a given // position (x, y) in m x n matrix, // boundaries of the matrix will not be crossed. double findProbability(int m, int n, int x,
int y, int N)
{ // boundary crossed
if (!isSafe(x, y, m, n))
return 0.0;
// N steps taken
if (N == 0)
return 1.0;
// Initialize result
double prob = 0.0;
// move up
prob += findProbability(m, n, x - 1,
y, N - 1) * 0.25;
// move right
prob += findProbability(m, n, x,
y + 1, N - 1) * 0.25;
// move down
prob += findProbability(m, n, x + 1,
y, N - 1) * 0.25;
// move left
prob += findProbability(m, n, x,
y - 1, N - 1) * 0.25;
return prob;
} // Driver code int main()
{ // matrix size
int m = 5, n = 5;
// coordinates of starting point
int i = 1, j = 1;
// Number of steps
int N = 2;
cout << "Probability is "
<< findProbability(m, n, i, j, N);
return 0;
} |
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Java
// Java program to find the probability // that we do not cross boundary // of a matrix after N moves. import java.io.*;
class GFG {
// check if (x, y) is valid // matrix coordinate static boolean isSafe(int x, int y,
int m, int n)
{ return (x >= 0 && x < m &&
y >= 0 && y < n);
} // Function to calculate probability // that after N moves from a given // position (x, y) in m x n matrix, // boundaries of the matrix will // not be crossed. static double findProbability(int m, int n,
int x, int y,
int N)
{ // boundary crossed
if (! isSafe(x, y, m, n))
return 0.0;
// N steps taken
if (N == 0)
return 1.0;
// Initialize result
double prob = 0.0;
// move up
prob += findProbability(m, n, x - 1,
y, N - 1) * 0.25;
// move right
prob += findProbability(m, n, x, y + 1,
N - 1) * 0.25;
// move down
prob += findProbability(m, n, x + 1,
y, N - 1) * 0.25;
// move left
prob += findProbability(m, n, x, y - 1,
N - 1) * 0.25;
return prob;
} // Driver code public static void main (String[] args)
{ // matrix size
int m = 5, n = 5;
// coordinates of starting point
int i = 1, j = 1;
// Number of steps
int N = 2;
System.out.println("Probability is " +
findProbability(m, n, i,
j, N));
} } // This code is contributed by KRV. |
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Python3
# Python3 program to find the probability # that we do not cross boundary of a # matrix after N moves. # check if (x, y) is valid matrix coordinate def isSafe(x, y, m, n):
return (x >= 0 and x < m and
y >= 0 and y < n)
# Function to calculate probability # that after N moves from a given # position (x, y) in m x n matrix, # boundaries of the matrix will # not be crossed. def findProbability(m, n, x, y, N):
# boundary crossed
if (not isSafe(x, y, m, n)):
return 0.0
# N steps taken
if (N == 0):
return 1.0
# Initialize result
prob = 0.0
# move up
prob += findProbability(m, n, x - 1,
y, N - 1) * 0.25
# move right
prob += findProbability(m, n, x,
y + 1, N - 1) * 0.25
# move down
prob += findProbability(m, n, x + 1,
y, N - 1) * 0.25
# move left
prob += findProbability(m, n, x,
y - 1, N - 1) * 0.25
return prob
# Driver code if __name__ == '__main__':
# matrix size
m = 5
n = 5
# coordinates of starting po
i = 1
j = 1
# Number of steps
N = 2
print("Probability is",
findProbability(m, n, i, j, N))
# This code is contributed by PranchalK |
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C#
// C# program to find the probability // that we do not cross boundary // of a matrix after N moves. using System;
class GFG
{ // check if (x, y) is valid // matrix coordinate static bool isSafe(int x, int y,
int m, int n)
{ return (x >= 0 && x < m &&
y >= 0 && y < n);
} // Function to calculate probability // that after N moves from a given // position (x, y) in m x n matrix, // boundaries of the matrix will // not be crossed. static double findProbability(int m, int n,
int x, int y,
int N)
{ // boundary crossed
if (! isSafe(x, y, m, n))
return 0.0;
// N steps taken
if (N == 0)
return 1.0;
// Initialize result
double prob = 0.0;
// move up
prob += findProbability(m, n, x - 1,
y, N - 1) * 0.25;
// move right
prob += findProbability(m, n, x, y + 1,
N - 1) * 0.25;
// move down
prob += findProbability(m, n, x + 1,
y, N - 1) * 0.25;
// move left
prob += findProbability(m, n, x, y - 1,
N - 1) * 0.25;
return prob;
} // Driver code public static void Main ()
{ // matrix size
int m = 5, n = 5;
// coordinates of starting point
int i = 1, j = 1;
// Number of steps
int N = 2;
Console.Write("Probability is " +
findProbability(m, n, i,
j, N));
} } // This code is contributed by nitin mittal. |
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PHP
<?php // PHP program to find the probability // that we do not cross boundary of a // matrix after N moves. // check if (x, y) is valid // matrix coordinate function isSafe($x, $y, $m, $n)
{ return ($x >= 0 && $x < $m &&
$y >= 0 && $y < $n);
} // Function to calculate probability // that after N moves from a given // position (x, y) in m x n matrix, // boundaries of the matrix will // not be crossed. function findProbability($m, $n, $x,
$y, $N)
{ // boundary crossed
if (!isSafe($x, $y, $m, $n))
return 0.0;
// N steps taken
if ($N == 0)
return 1.0;
// Initialize result
$prob = 0.0;
// move up
$prob += findProbability($m, $n, $x - 1,
$y, $N - 1) * 0.25;
// move right
$prob += findProbability($m, $n, $x,
$y + 1, $N - 1) * 0.25;
// move down
$prob += findProbability($m, $n, $x + 1,
$y, $N - 1) * 0.25;
// move left
$prob += findProbability($m, $n, $x,
$y - 1, $N - 1) * 0.25;
return $prob;
} // Driver code // matrix size $m = 5; $n = 5;
// coordinates of starting point $i = 1; $j = 1;
// Number of steps $N = 2;
echo "Probability is ",
findProbability($m, $n, $i, $j, $N);
// This code is contributed by nitin mittal. ?> |
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Output :
Probability is 0.875
This article is contributed by Aditya Goel. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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