1 A roller-coaster ride has a safety system to detect faults on the track.
- State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day.
Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
- Find the probability that on a randomly chosen day there are
(A) no faults,
(B) at least 2 faults. - Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults.
There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
- State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
[0pt] - The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]