Compute Poisson probabilities for 3.2 arrivals per hour with math.exp and math.factorial, and rescale the rate to a 15-minute window for P(no arrivals).

Answer

import math def pois(x, lam): return math.exp(-lam) * lam ** x / math.factorial(x) lam_hour = 3.2 lam_quarter = lam_hour * 15 / 60 print(f"P(X=2)={pois(2, lam_hour):.4f} P(X<=1)={pois(0, lam_hour) + pois(1, lam_hour):.4f} P(0 in 15 min)={pois(0, lam_quarter):.4f}")

IBS1 §4.4 (Poisson, proportional rescaling of the rate), IS1 §4.6, CC BY 4.0 — shape only; original dataset