Statistic and Probability Q3

Question 1
In a production process, a total number of 50 items were produced by machines M1, M2 and M3. Of those items being produced, 27 were made by machine M1, 13 were made by M2, and 10 were made by M3.
The three machines work independently, however, they do not work perfectly.
From the past experience, 9% of the items produced by M1 are defective, 7% of the items produced by M2 are defective, and 4% of the items produced by M3 are defective.
(a) Given that a randomly selected item is non-defective, what is the probability that it is produced by machine M2?
(b) If two items which are not produced by machine M3 are selected at random without replacement, what is the probability that at least one of them is non-defective?

Question 2

A machine produces vitamin C tablets, the diameters of which are normally distributed with mean 10mm and standard deviation 0.1mm. A tablet is acceptable if its diameter lies between 9.81mm and 10.19mm. To pass a quality test, at least 90% of the tablets in a pack should be acceptable.

(a) Determine the probability that a tablet is acceptable.
(b) What is the probability that a pack of 20 tablets will pass the quality test?
(c) What is the probability that a pack of 200 tablets will pass the quality test?


Question 3

A factory produces a certain type of ball bearings whose outer diameters are normally distributed with mean 52 mm and standard deviation 0.05 mm. A ball bearing of this type is suitable for manufacturing certain electric motors if its outer diameter is between 51.92 mm and 52.06 mm. Using normal approximation to binomial, find the minimum number of ball bearings that have to be produced so that the probability of having at least 500 ball bearings suitable for manufacturing the electric motors is at least 0.95.