Atsižvelgiant į stačiakampę matricą, mes galime pereiti iš dabartinės ląstelės 4 kryptimis su vienoda tikimybe. 4 kryptys yra dešinės kairės viršaus arba apačios. Apskaičiuokite tikimybę, kad po n judės iš nurodytos padėties (i j) matricoje, bet kuriuo metu neperžengsime matricos ribų.
java vizualizatorius
Mes labai rekomenduojame sumažinti savo naršyklę ir pirmiausia išbandyti tai patys.
Idėja yra atlikti kažką panašaus į DFS. Mes rekursyviai einame kiekvienoje iš 4 leidžiamų krypčių, o kiekvienai sutinkamai ląstelei apskaičiuojame reikiamą tikimybę vienu mažesniu judesiu. Kadangi kiekviena kryptis turi vienodą tikimybę, kiekviena kryptimi prisidės prie 1/4 visos tos ląstelės tikimybės, t. Y. 0,25. Mes grąžiname 0, jei išeiname už matricos ribų ir grąžiname 1, jei n žingsniai bus atlikti neperžengdami matricos ribų.
Žemiau yra aukščiau pateiktos idėjos įgyvendinimas:
C++/// C++ program to find the probability // that we do not cross boundary of a // matrix after N moves. #include
/// C program to find the probability // that we do not cross boundary of a // matrix after N moves. #include
// 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 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# 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 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. ?> <script> // JavaScript 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 let 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 let m = 5 n = 5; // coordinates of starting point let i = 1 j = 1; // Number of steps let N = 2; document.write( 'Probability is ' +findProbability(m n i j N)); </script>
Išvestis
Probability is 0.875