Probability of reaching a point with 2 or 3 steps at a time

A person starts walking from position X = 0, find the probability to reach exactly on X = N if she can only take either 2 steps or 3 steps. Probability for step length 2 is given i.e. P, probability for step length 3 is 1 – P.

Examples :

Input : N = 5, P = 0.20
Output : 0.32
Explanation :-
There are two ways to reach 5.
2+3 with probability = 0.2 * 0.8 = 0.16
3+2 with probability = 0.8 * 0.2 = 0.16
So, total probability = 0.32.



It is a simple dynamic programming problem. It is simple extension of this problem :- count-ofdifferent-ways-express-n-sum-1-3-4

Below is the implementation of the above approach.

C++

// CPP Program to find probability to 
// reach N with P probability to take
// 2 steps (1-P) to take 3 steps
#include <bits/stdc++.h>
using namespace std;
  
// Returns probability to reach N
float find_prob(int N, float P)
{
    double dp[N + 1];
    dp[0] = 1;
    dp[1] = 0;
    dp[2] = P;
    dp[3] = 1 - P;
    for (int i = 4; i <= N; ++i)
        dp[i] = (P)*dp[i - 2] + (1 - P) * dp[i - 3];
  
    return dp[N];
}
  
// Driver code
int main()
{
    int n = 5;
    float p = 0.2;
    cout << find_prob(n, p);
    return 0;
}

Java

// Java Program to find probability to 
// reach N with P probability to take
// 2 steps (1-P) to take 3 steps
import java.io.*;
  
class GFG {
      
    // Returns probability to reach N
    static float find_prob(int N, float P)
    {
        double dp[] = new double[N + 1];
        dp[0] = 1;
        dp[1] = 0;
        dp[2] = P;
        dp[3] = 1 - P;
      
        for (int i = 4; i <= N; ++i)
          dp[i] = (P) * dp[i - 2] +
                        (1 - P) * dp[i - 3];
      
        return ((float)(dp[N]));
    }
      
    // Driver code
    public static void main(String args[])
    {
        int n = 5;
        float p = 0.2f;
        System.out.printf("%.2f",find_prob(n, p));
    }
}
  
  
/* This code is contributed by Nikita Tiwari.*/

Python3

# Python 3 Program to find 
# probability to reach N with
# P probability to take 2 
# steps (1-P) to take 3 steps
  
# Returns probability to reach N
def find_prob(N, P) :
      
    dp =[0] * (n + 1)
    dp[0] = 1
    dp[1] = 0
    dp[2] = P
    dp[3] = 1 - P
      
    for i in range(4, N + 1) :
        dp[i] = (P) * dp[i - 2] + (1 - P) * dp[i - 3]
  
    return dp[N]
  
# Driver code
n = 5
p = 0.2
print(round(find_prob(n, p), 2))
  
# This code is contributed by Nikita Tiwari.

C#

// C# Program to find probability to 
// reach N with P probability to take
// 2 steps (1-P) to take 3 steps
using System;
  
class GFG {
      
    // Returns probability to reach N
    static float find_prob(int N, float P)
    {
        double []dp = new double[N + 1];
        dp[0] = 1;
        dp[1] = 0;
        dp[2] = P;
        dp[3] = 1 - P;
      
        for (int i = 4; i <= N; ++i)
        dp[i] = (P) * dp[i - 2] +
                (1 - P) * dp[i - 3];
      
        return ((float)(dp[N]));
    }
      
    // Driver code
    public static void Main()
    {
        int n = 5;
        float p = 0.2f;
        Console.WriteLine(find_prob(n, p));
    }
}
  
  
/* This code is contributed by vt_m.*/

PHP

<?php
// PHP Program to find probability to 
// reach N with P probability to take
// 2 steps (1-P) to take 3 steps
  
// Returns probability to reach N
function find_prob($N, $P)
{
    $dp;
    $dp[0] = 1;
    $dp[1] = 0;
    $dp[2] = $P;
    $dp[3] = 1 - $P;
    for ($i = 4; $i <= $N; ++$i)
        $dp[$i] = ($P) * $dp[$i - 2] + 
                  (1 - $P) * $dp[$i - 3];
  
    return $dp[$N];
}
  
// Driver code
$n = 5;
$p = 0.2;
echo find_prob($n, $p);
  
// This code is contributed by mits.
?>


Output :

0.32


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