Actuaries A, B, C always arrive late to their company: in fact, the boss already knows that A is late 10% of the days, B is late 20% of the days and that C, the most unfulfilled, arrives late one in every three days, that is, 33% of the days (they have not fired actuary C, although they have tried many times, because only he is an expert in that area that he handles and that no one knows better than him; They have been very patient). If the three actuaries arrive each day to work in a mutually independent manner, determine the probability that, tomorrow, AT LEAST ONE of them will be late.

To find the probability that at least one of the three actuaries (A, B, or C) will be late tomorrow, we can use the complementary probability approach. We'll find the probability that none of them are late and then subtract that probability from 1 to get the probability that at least one is late. Let: - P(A) be the probability that Actuary A is late tomorrow, which is 0.10 (10%). - P(B) be the probability that Actuary B is late tomorrow, which is 0.20 (20%). - P(C) be the probability that Actuary C is late tomorrow, which is 0.33 (33%). Since they arrive independently, the probability that none of them is late tomorrow (all arrive on time) is: P(None late) = P(A is on time) * P(B is on time) * P(C is on time) P(None late) = (1 - P(A)) * (1 - P(B)) * (1 - P(C)) P(None late) = (1 - 0.10) * (1 - 0.20) * (1 - 0.33) P(None late) = 0.90 * 0.80 * 0.67 Now, calculate P(None late): P(None late) ≈ 0.4824 Now, we want the probability that at least one of them is late, which is the complement of P(None late): P(At least one late) = 1 - P(None late) P(At least one late) = 1 - 0.4824 P(At least one late) ≈ 0.5176 So, the probability that at least one of the three actuaries will be late tomorrow is approximately 0.5176 or 51.76%.