Trajectories, random walk and binomial convergence

Statement

A server receives weekly updates for n weeks. Each week there are m independent attackers; each attacker has probability p of succeeding. If at least one succeeds, the week is a breach and counts as -1; if none succeed the week is secure and counts as +1.

\[ \displaystyle q = (1-p)^m . \] \[ \displaystyle K\sim\mathrm{Binom}(n,q), \qquad S_n = 2K - n . \]

Interactive simulator

Weeks (n)

Attackers per week (m)

Single attacker probability (p)

Simulations (sims)

Instructions

Choose parameters and click Run simulation.
The app will plot some sample trajectories, the empirical mean, ±sd bands and the theoretical expectation.

Numerical results

Score S_n	Count (sim)	Prob (sim)	Prob (theor)

Metrics

Run the simulation to see metrics here (final mean, variance, Chi², KL divergence).

Convergence sweep

Run a quick sweep to observe how Chi² behaves when varying n or the number of simulations.

n from

step

Interpretation & final notes

The empirical mean approaches E[S_k]=k(2q-1). Increasing sims and n makes the empirical distribution converge to the theoretical one; changing m modifies q and shifts the regime.

References

W. Feller, An Introduction to Probability Theory and Its Applications, Vol.1.
S. Ross, Introduction to Probability Models.
G. Grimmett & D. Stirzaker, Probability and Random Processes.