Probability of collision between two trucks

Given two strings S and T, where S represents the first lane in which vehicles move from left to right and T represents the second lane in which vehicles move from right to left. Vehicles can be either B (bike), C (car), or T (truck). The task is to find the probability of collision between two trucks.

Examples:

Input: S = “TCCBCTTB”, T = “BTCCBBTT”
Output: 0.194444
Explanation:
Total collision = 7
Total accident = 36
Therefore, the probability can be calculated by 7/36 = 0.19444.

Input: S = “BTT”, T = “BTCBT”
Output: 0.25000

Illustration:

S = "TCCBCTTB", T = "BTCCBBTT"

Possible cases   | Accidents | Collision
-----------------------------------------       
TCCBCTTB         |           |
BTCCBBTT         |     8     |   1
                 |           |  
 TCCBCTTB        |           |
BTCCBBTT         |     7     |   3
                 |           |
  TCCBCTTB       |           |
BTCCBBTT         |     6     |   1
                 |           |
   TCCBCTTB      |           |
BTCCBBTT         |     5     |   0
                 |           |
    TCCBCTTB     |           |
BTCCBBTT         |     4     |   0
                 |           |
     TCCBCTTB    |           |
BTCCBBTT         |     3     |   0
                 |           |
      TCCBCTTB   |           |
BTCCBBTT         |     2     |   1
                 |           |
       TCCBCTTB  |           |
BTCCBBTT         |     1     |   1

Total number of accidents: 8+7+6+5+4+3+2+1=36
Total number of collision: 1+3+1+0+0+0+1+1=7
Probability: 7/36=0.19444

Approach: Follow the steps below to solve the problem:

• Find the total number of favorable outcomes as the total number of collisions(accidents between trucks) and the total number of possible outcomes(total number of collisions) as the total number of accidents.
• Initialize a variable answer equals to 0 to stores the count of collisions.
• Count the number of trucks in the string T and store it in a variable count.
• Iterate over the characters of the strings S and T simultaneously:
  • If S[i] is equal to ‘T’, increment answer by count.
  • If T[i] is equal to ‘T’, decrement the count by 1.
• Now, calculate the total number of possible outcomes (total number of accidents). It is the sum of all the length of overlapping if keep shifting string a towards right or string b towards left by one unit.
• Let the length of string be N and string b be M. Then, the total number of overlapping will be:
  • If N > M then, it will be the sum of first M natural numbers i.e., M*(M + 1)/2.
  • Otherwise, it will be N*(N + 1)/2 + (M – N)*N.
• Find the probability as the ratio of the count of the collisions and the count of the accidents.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to calculate total
// number of accidents
double count_of_accident(string a,
                         string b)
{
    // String size
    int n = a.size(), m = b.size();

    if (n > m)
        return (m * (m + 1)) / 2;
    else
        return (n * (n + 1))
                   / 2
               + (m - n) * n;
}

// Function to calculate count
// of all possible collision
double count_of_collision(string a,
                          string b)
{
    int n = a.size(), m = b.size();

    // Stores the count of collisions
    int answer = 0;

    // Total number of truck in lane b
    int count_of_truck_in_lane_b = 0;
    for (int i = 0; i < m; i++)
        if (b[i] == 'T')
            count_of_truck_in_lane_b++;

    // Count total number of collisions
    // while traversing the string a
    for (int i = 0; i < n && i < m; i++) {
        if (a[i] == 'T')
            answer
                += count_of_truck_in_lane_b;

        if (b[i] == 'T')
            count_of_truck_in_lane_b--;
    }
    return answer;
}

// Function to calculate the
// probability of collisions
double findProbability(string a,
                       string b)
{
    // Evaluate total outcome that is
    // all the possible accident
    double total_outcome
        = count_of_accident(a, b);

    // Evaluate favourable outcome i.e.,
    // count of collision of trucks
    double favourable_outcome
        = count_of_collision(a, b);

    // Print desired probability
    cout << favourable_outcome
                / total_outcome;
}

// Driver Code
int main()
{
    string S = "TCCBCTTB", T = "BTCCBBTT";

    // Function Call
    findProbability(S, T);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;

class GFG
{

// Function to calculate total
// number of accidents
static int count_of_accident(String a,
                         String b)
{
    // String size
    int n = a.length(), m = b.length();

    if (n > m)
        return (m * (m + 1)) / 2;
    else
        return (n * (n + 1))
                   / 2
               + (m - n) * n;
}

// Function to calculate count
// of all possible collision
static double count_of_collision(String a,
                          String b)
{
    int n = a.length(), m = b.length();

    // Stores the count of collisions
    double answer = 0;

    // Total number of truck in lane b
    int count_of_truck_in_lane_b = 0;
    for (int i = 0; i < m; i++)
        if (b.charAt(i) == 'T')
            count_of_truck_in_lane_b++;

    // Count total number of collisions
    // while traversing the String a
    for (int i = 0; i < n && i < m; i++)
    {
        if (a.charAt(i) == 'T')
            answer
                += count_of_truck_in_lane_b;

        if (b.charAt(i) == 'T')
            count_of_truck_in_lane_b--;
    }
    return answer;
}

// Function to calculate the
// probability of collisions
static void findProbability(String a,
                       String b)
{
    // Evaluate total outcome that is
    // all the possible accident
    int total_outcome
        = count_of_accident(a, b);

    // Evaluate favourable outcome i.e.,
    // count of collision of trucks
    double favourable_outcome
        = count_of_collision(a, b);

    // Print desired probability
    System.out.printf("%4f",favourable_outcome
                / total_outcome);
}

// Driver Code
public static void main(String[] args)
{
    String S = "TCCBCTTB", T = "BTCCBBTT";

    // Function Call
    findProbability(S, T);
}
}

// This code is contributed by 29AjayKumar

Python3

# Python3 program for the above approach

# Function to calculate total
# number of accidents
def count_of_accident(a, b):

    n = len(a)
    m = len(b)

    if (n > m):
        return (m * (m + 1)) / 2
    else:
        return ((n * (n + 1)) / 2 +
                (m - n) * n)

# Function to calculate count
# of all possible collision
def count_of_collision(a, b):

    # Size of string
    n = len(a)
    m = len(b)

    # Stores the count of collisions
    answer = 0

    # Total number of truck in lane b
    count_of_truck_in_lane_b = 0

    for i in range(0, m):
        if (b[i] == 'T'):
            count_of_truck_in_lane_b += 1

    # Count total number of collisions
    # while traversing the string a
    i = 0
    while (i < m and i < n):
        if (a[i] == 'T'):
            answer += count_of_truck_in_lane_b
        if (b[i] == 'T'):
            count_of_truck_in_lane_b -= 1

        i += 1

    return answer

# Function to calculate the
# probability of collisions
def findProbability(a, b):

    # Evaluate total outcome that is
    # all the possible accident
    total_outcome = count_of_accident(a, b);

    # Evaluate favourable outcome i.e.,
    # count of collision of trucks
    favourable_outcome = count_of_collision(a, b);

    # Print desired probability
    print(favourable_outcome / total_outcome)

# Driver Code
if __name__ == "__main__" :

    S = "TCCBCTTB"
    T = "BTCCBBTT"

    # Function Call
    findProbability(S, T)

# This code is contributed by Virusbuddah_

C#

// C# program for the above approach
using System;

class GFG
{

// Function to calculate total
// number of accidents
static int count_of_accident(String a,
                         String b)
{
    // String size
    int n = a.Length, m = b.Length;

    if (n > m)
        return (m * (m + 1)) / 2;
    else
        return (n * (n + 1))
                   / 2
               + (m - n) * n;
}

// Function to calculate count
// of all possible collision
static double count_of_collision(String a,
                          String b)
{
    int n = a.Length, m = b.Length;

    // Stores the count of collisions
    double answer = 0;

    // Total number of truck in lane b
    int count_of_truck_in_lane_b = 0;
    for (int i = 0; i < m; i++)
        if (b[i] == 'T')
            count_of_truck_in_lane_b++;

    // Count total number of collisions
    // while traversing the String a
    for (int i = 0; i < n && i < m; i++)
    {
        if (a[i] == 'T')
            answer
                += count_of_truck_in_lane_b;

        if (b[i] == 'T')
            count_of_truck_in_lane_b--;
    }
    return answer;
}

// Function to calculate the
// probability of collisions
static void findProbability(String a,
                       String b)
{
    // Evaluate total outcome that is
    // all the possible accident
    int total_outcome
        = count_of_accident(a, b);

    // Evaluate favourable outcome i.e.,
    // count of collision of trucks
    double favourable_outcome
        = count_of_collision(a, b);

    // Print desired probability
    Console.Write("{0:F4}", favourable_outcome
                / total_outcome);
}

// Driver Code
public static void Main(String[] args)
{
    String S = "TCCBCTTB", T = "BTCCBBTT";

    // Function Call
    findProbability(S, T);
}
}


// This code is contributed by sapnasingh4991

Javascript

<script>

// Javascript program to implement
// the above approach

// Function to calculate total
// number of accidents
function count_of_accident(a, b)
{
    // String size
    let n = a.length, m = b.length;

    if (n > m)
        return (m * (m + 1)) / 2;
    else
        return (n * (n + 1))
                   / 2
               + (m - n) * n;
}

// Function to calculate count
// of all possible collision
function count_of_collision(a, b)
{
    let n = a.length, m = b.length;

    // Stores the count of collisions
    let answer = 0;

    // Total number of truck in lane b
    let count_of_truck_in_lane_b = 0;
    for (let i = 0; i < m; i++)
        if (b[i] == 'T')
            count_of_truck_in_lane_b++;

    // Count total number of collisions
    // while traversing the String a
    for (let i = 0; i < n && i < m; i++)
    {
        if (a[i] == 'T')
            answer
                += count_of_truck_in_lane_b;

        if (b[i] == 'T')
            count_of_truck_in_lane_b--;
    }
    return answer;
}

// Function to calculate the
// probability of collisions
function findProbability(a, b)
{
    // Evaluate total outcome that is
    // all the possible accident
    let total_outcome
        = count_of_accident(a, b);

    // Evaluate favourable outcome i.e.,
    // count of collision of trucks
    let favourable_outcome
        = count_of_collision(a, b);

    // Print desired probability
    document.write( favourable_outcome
                / total_outcome);
}

    // Driver Code

    let S = "TCCBCTTB", T = "BTCCBBTT";

    // Function Call
    findProbability(S, T);

// This code is contributed by souravghosh0416.
</script>

Output:

0.194444

Time Complexity: O(N)
Auxiliary Space: O(1)