Een matrixkansvraag
Gegeven een rechthoekige matrix kunnen we met gelijke waarschijnlijkheid in 4 richtingen van de huidige cel gaan. De 4 aanwijzingen zijn rechts links of onderkant. Bereken de waarschijnlijkheid dat na n van een bepaalde positie (i j) in de matrix beweegt, we op geen enkel moment grenzen van de matrix overschrijden.
We raden u ten zeerste aan om uw browser te minimaliseren en dit eerst zelf te proberen.
Het idee is om iets vergelijkbaars uit te voeren met DFS. We doorkruisen recursief in elk van de 4 toegestane richting en voor elke aangetroffen cel berekenen we de vereiste waarschijnlijkheid met één beweging minder. Aangezien elke richting gelijke waarschijnlijkheid heeft, zal elke richting bijdragen aan 1/4 van de totale waarschijnlijkheid van die cel, d.w.z. 0,25. We retourneren 0 Als we buiten de matrix stappen en 1 retourneren als n stappen zijn voltooid zonder matrixgrenzen te overschrijden.
Hieronder is de implementatie van bovenstaande idee:
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 ; }
C /// C program to find the probability // that we do not cross boundary of a // matrix after N moves. #include #include // 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 ; printf ( 'Probability is %f' findProbability ( m n i j N )); return 0 ; } // This code is contributed by kothavvsaakash.
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 // 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. ?>
JavaScript < 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>
Uitvoer
Probability is 0.875
