The coding part was fine, nested loops, count survivors, average across trials.
First, derive the exact probability analytically: each particle survives 10 days with probability (1/2)^10 = 1/1024, so the probability at least one of 100 particles survives is 1 - (1 - 1/1024)^100. Then implement a Monte Carlo simulation with 100,000 trials, each simulating 10 days for 100 particles, and compare the estimated probability to the exact value, discussing statistical error and convergence.
Pro tip: Mention that the exact answer can be computed efficiently using logarithms to avoid numerical underflow, and that the Monte Carlo standard error can be estimated as sqrt(p*(1-p)/N) to quantify uncertainty.
Compute the exact probability that at least one particle survives after 10 days. Each particle independently survives with probability (1/2)^10 = 1/1024, so the probability that none survive is (1 - 1/1024)^100, and the desired probability is 1 minus that.
Plan the Monte Carlo simulation: for each of 100,000 trials, simulate 100 particles over 10 days. For each particle, at each day, draw a uniform random number and decay if it is less than 0.5. Track whether at least one particle survives the full 10 days.
Implement the simulation in code, using vectorized operations or early termination when all particles have decayed to speed up execution. Ensure the random number generator is seeded for reproducibility.
Compute the estimated probability from the simulation and compare it to the exact analytical value. Calculate the standard error of the estimate and discuss whether the difference is within expected statistical fluctuations.
Comment on how the estimate converges as the number of trials increases, and mention potential extensions such as varying the decay probability or number of particles to test the simulation's robustness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.