Probability

(a)
Two production lines A and B produce respectively 30% and 70% of the total number of steel rods in a factory. The lengths of rods manufactured by each line are normally distributed with the following means and standard deviations.

Population mean of length of Line A is 23 cm
Population standard deviation of length of Line A is 2.5 cm

Population mean of length of Line B is 21 cm
Population standard deviation of length of Line B is 1.5 cm

Steel rods are classified as “below standard” if the lengths are shorter than 20 cm. Find the probabilities of the following events. Correct your answers to 4 decimal places.
(i)
A randomly selected rod from line A is below standard.
(ii)
A randomly selected rod from line B is below standard.
(Iii)
A randomly selected rod from the factory is below standard.
(iv)
A randomly selected rod is produced by line A given that it is below standard.