Mathematics
Mathematics
Problem
Three independent machines , and manufacture identical bolts. On any given day, the probabilities that the machines are working satisfactorily are , and respectively. (i) Find the probability that all three machines work satisfactorily on that day. (ii) Calculate the probability that exactly two of the machines work satisfactorily. (iii) Determine the probability that at least one machine is out of order.Intuitive explanation
This problem is about independence and complementary events in probability. Since the machines operate independently, you multiply probabilities for "all working together," and you use the complement rule (1 − P(event)) to flip between "working" and "broken." For part (ii), "exactly two work" means you pick which two succeed, calculate that scenario, then add up all such cases. Part (iii) flips the question: instead of calculating many cases where at least one fails, find the probability all three work, then subtract from 1.
Intuitive explanation
This problem is about independence and complementary events in probability. Since the machines operate independently, you multiply probabilities for "all working together," and you use the complement rule (1 − P(event)) to flip between "working" and "broken." For part (ii), "exactly two work" means you pick which two succeed, calculate that scenario, then add up all such cases. Part (iii) flips the question: instead of calculating many cases where at least one fails, find the probability all three work, then subtract from 1.