We are talking about server security over a period of n weeks. In this assignment we apply fundamental principles of probability theory to a practical, real-world context: the security history of a server. Our approach centers on developing a computational model that simulates risk over time.
We analyze server security over a period of nn weeks by applying probability theory to a realistic scenario of cyber risk. The approach models each week as an independent Bernoulli trial, where the server either remains secure (+1) or is breached (–1). With m attackers each week, the probability that no one succeeds (and thus the server stays secure) is (1−p)^m.
In this model, an attacker’s chance of successfully breaching the server increases as the individual breach probability prises, or as the number of attackers m increases. The weekly probability that the server remains secure is (1−p)m(1−p)m; therefore, even if p is small, having many attackers greatly increases the overall risk of a breach. Conversely, reducing either p (through stronger defenses) or m (fewer simultaneous threats) improves the likelihood of staying secure each week. This relationship illustrates how collective risk compounds with the number of attackers and highlights the nonlinear nature of security probability.
This random-walk analogy transforms the weekly outcomes into a sequence of steps, allowing us to simulate many trajectories and compute cumulative security scores. The final number of secure weeks k out of n follows a Binomial distribution B(n, (1−p)^m). Each possible score corresponds to a term in Pascal’s Triangle, connecting the model to classic combinatorial structures and the Binomial Theorem.
where (n k) represents the number of distinct trajectories (paths in the random walk) leading to exactly k secure and n−k breached weeks.
When (1−p)^m=0.5, the process becomes a symmetric random walk, and the coefficients form the rows of Pascal’s Triangle. By comparing thousands of simulated trajectories with the theoretical Binomial probabilities, we show that the empirical frequencies converge to the expected Binomial law, illustrating the Law of Large Numbers and the emergence of normality predicted by the Central Limit Theorem.
In the simulation, each week is modeled as an independent Bernoulli trial where the server either remains secure (+1) or is breached (–1). The probability of staying secure is (1−p)m, meaning no attacker (out of mm) succeeds. This framework covers all possible weekly outcomes and all possible total scores from –n to +n. The empirical results from thousands of simulated trajectories are compared with the theoretical Binomial distribution Bin(n,(1−p)m), showing that the simulated frequencies converge to the expected probabilities as the number of trajectories increases. The model assumes independent weeks and identical attackers, which is appropriate for capturing the essential random-walk behavior of the system’s cumulative security score.
This work effectively bridges theoretical mathematics and computational practice, providing a robust visualization of how fundamental laws, such as the Law of Large Numbers and principles related to the Central Limit Theorem, govern seemingly random processes. We will quantify this convergence by comparing our simulation results with the probabilities derived from the binomial coefficients.