A matrix probability question

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 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; }

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.

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

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.

PHP

 = 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.  ?>

Output :

Probability is 0.875

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