Calculating the Probability after One Step

Oct 13, 2025 by windailys.com 98 views

Introduction

In the realm of probability and statistics, path-counting problems are a crucial aspect of understanding the behavior of random processes. These problems involve counting the number of paths or trajectories that a random process can take, often in a discrete or finite state space. In this article, we will explore a specific type of path-counting problem related to a triangular room division, where a small ball moves from one room to an adjacent room with equal probability.

Problem Description

As shown in the figure below, a triangle is divided into 9 rooms. Two rooms with a common side are called adjacent rooms. A small ball moves from one room to an adjacent room with equal probability each time. The goal is to find the probability of the ball ending up in a specific room after a certain number of steps.

Figure: Triangular Room Division

  1 | 2 | 3
  ---------
 4 | 5 | 6
  ---------
 7 | 8 | 9

Mathematical Formulation

Let's denote the rooms as $R_1, R_2, ..., R_9$ and the number of steps as $n$ . We can represent the state of the ball at each step as a vector $(x, y)$ , where $x$ and $y$ are the coordinates of the room the ball is currently in. The transition probability from one room to an adjacent room is given by:

P(R_i \rightarrow R_j) = \begin{cases} \frac{1}{3} & \text{if } R_i \text{ and } R_j \text{ are adjacent} \\ 0 & \text{otherwise} \end{cases}

The probability of the ball ending up in a specific room $R_k$ after $n$ steps can be calculated using the formula:

P(R_k \text{ at time } n) = \sum_{i=1}^{9} P(R_i \text{ at time } n-1) \cdot P(R_i \rightarrow R_k)

Solution Approach

To solve this problem, we can use a recursive approach. We can start by calculating the probability of the ball ending up in each room after one step, and then use this information to calculate the probability after two steps, and so on.

Calculating the Probability after One Step

Let's denote the probability of the ball ending up in room $R_i$ after one step as $P(R_i \text{ at time } 1)$ . We can calculate this probability using the transition probability formula:

P(R_i \text{ at time } 1) = \sum_{j=1}^{9} P(R_j \text{ at time } 0) \cdot P(R_j \rightarrow R_i)

Since the ball starts in room $R_5$ , we have:

P(R_i \text{ at time } 1) = \begin{cases} \frac{1}{3} & \text{if } i = 1, 2, 3, 4, 6, 7, 8, 9 \\ 0 & \text{otherwise} \end{cases}

Calculating the Probability after Two Steps

Now that we have the probability of the ball ending up in each room after one step, we can calculate the probability after two steps. We can use the formula:

P(R_k \text{ at time } 2) = \sum_{i=1}^{9} P(R_i \text{ at time } 1) \cdot P(R_i \rightarrow R_k)

Substituting the values, we get:

P(R_k \text{ at time } 2) = \begin{cases} \frac{1}{9} & \text{if } k = 1, 2, 3, 4, 6, 7, 8, 9 \\ \frac{2}{9} & \text{if } k = 5 \end{cases}

Generalizing the Solution

We can generalize the solution to find the probability of the ball ending up in a specific room after $n$ steps. We can use the formula:

P(R_k \text{ at time } n) = \sum_{i=1}^{9} P(R_i \text{ at time } n-1) \cdot P(R_i \rightarrow R_k)

Substituting the values, we get:

P(R_k \text{ at time } n) = \begin{cases} \frac{1}{9} & \text{if } k = 1, 2, 3, 4, 6, 7, 8, 9 \\ \frac{2}{9} & \text{if } k = 5 \end{cases}

Conclusion

In this article, we explored a specific type of path-counting problem related to a triangular room division. We formulated the problem mathematically and used a recursive approach to solve it. We found that the probability of the ball ending up in a specific room after $n$ steps is given by a simple formula. This problem has applications in various fields, including probability theory, statistics, and computer science.

Future Work

There are several directions for future work. One possible extension is to consider a more general transition probability matrix, where the probability of moving from one room to an adjacent room is not necessarily equal. Another possible extension is to consider a larger or smaller triangular room division, or even a different shape of room division.

References

[1] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and random processes. Oxford University Press.
[2] Feller, W. (1968). An introduction to probability theory and its applications. John Wiley & Sons.

Appendix

The following is a Python code snippet that implements the solution:

import numpy as np
def calculate_probability(n):
# Initialize the probability matrix
P = np.zeros((9, 9))
# Calculate the probability after one step
for i in range(9):
    for j in range(9):
        if i == j:
            P[i, j] = 1/3
        elif abs(i-j) == 1:
            P[i, j] = 1/3
        else:
            P[i, j] = 0

# Calculate the probability after n steps
for _ in range(n-1):
    P = np.dot(P, P)

return P

n = 10
P = calculate_probability(n)
print(P)

Introduction

In our previous article, we explored a specific type of path-counting problem related to a triangular room division. We formulated the problem mathematically and used a recursive approach to solve it. In this article, we will answer some of the most frequently asked questions related to this problem.

Q: What is the probability of the ball ending up in a specific room after n steps?

A: The probability of the ball ending up in a specific room after n steps is given by the formula:

P(R_k \text{ at time } n) = \begin{cases} \frac{1}{9} & \text{if } k = 1, 2, 3, 4, 6, 7, 8, 9 \\ \frac{2}{9} & \text{if } k = 5 \end{cases}

Q: How does the probability change as the number of steps increases?

A: As the number of steps increases, the probability of the ball ending up in a specific room remains the same. This is because the transition probability matrix is symmetric, and the probability of moving from one room to an adjacent room is equal.

Q: Can we generalize the solution to a larger or smaller triangular room division?

A: Yes, we can generalize the solution to a larger or smaller triangular room division. The key is to calculate the probability matrix for the new room division and then use the recursive approach to solve the problem.

Q: How does the transition probability matrix affect the probability of the ball ending up in a specific room?

A: The transition probability matrix affects the probability of the ball ending up in a specific room by determining the probability of moving from one room to an adjacent room. If the transition probability matrix is symmetric, the probability of the ball ending up in a specific room remains the same. However, if the transition probability matrix is not symmetric, the probability of the ball ending up in a specific room will change.

Q: Can we use this problem to model real-world scenarios?

A: Yes, we can use this problem to model real-world scenarios such as:

A random walk on a triangular lattice
A Markov chain with a triangular transition probability matrix
A problem in computer science such as finding the shortest path in a graph

Q: How can we extend this problem to include more complex scenarios?

A: We can extend this problem to include more complex scenarios by:

Introducing more rooms or a larger triangular room division
Changing the transition probability matrix to include more complex rules
Adding more constraints or objectives to the problem

Q: What are some of the applications of this problem?

A: Some of the applications of this problem include:

Probability theory and statistics
Computer science and graph theory
Physics and random walks
Engineering and optimization problems

Conclusion

In this article, we answered some of the most frequently asked questions related to the probability and path-counting problem related to a triangular room division. We hope that this article has provided a better understanding of the problem and its applications.

Future Work

References

[1] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and random processes. Oxford University Press.
[2] Feller, W. (1968). An introduction to probability theory and its applications. John Wiley & Sons.

Appendix

The following is a Python code snippet that implements the solution:

import numpy as np
def calculate_probability(n):
# Initialize the probability matrix
P = np.zeros((9, 9))
# Calculate the probability after one step
for i in range(9):
    for j in range(9):
        if i == j:
            P[i, j] = 1/3
        elif abs(i-j) == 1:
            P[i, j] = 1/3
        else:
            P[i, j] = 0

# Calculate the probability after n steps
for _ in range(n-1):
    P = np.dot(P, P)

return P

n = 10
P = calculate_probability(n)
print(P)

This code snippet calculates the probability of the ball ending up in each room after n steps. It uses a recursive approach to calculate the probability matrix, and then prints the resulting matrix.