3.

A. Machine A produces metal pieces that are cylindrical in shape. The diameter of the metal pieces are normally distributed with mean 3.6 centimeters and a standard deviation of 0.3 centimeter. i. Find the probability that a randomly selected metal piece produced by Machine A has the diameter below 3.85 centimeter. ii. Find the probability that 10 randomly selected metal pieces produced by Machine A has the mean diameter exceeds 3.45 centimeter. iii. Suppose that a metal piece is considered as defective if its diameter is below 3.45 centimeter or exceeds 3.85 centimeter. Find the probability that in a sample of 10 randomly selected metal pieces, there are less than 3 defective metal pieces.

B. Suppose that, on average, Anthony's smartphone receives 6 messages per hour. i. Determine the probability that at least 2 messages will be received in the next 30 minutes. ii. In a two hours exam, if 8 messages have been received, what is the probability that 2 messages have been received in the first 30 minutes?