A factory manufactures screws of which 3% have defects. We take at random 2000 screws.

a) Let X be the number of defective screws in the sample.

i) What is the law of X?

ii) If we want at least a 1950 screw without defect. What is the probability of having enough screws in good condition?

b) If no defective screws are found, quality control will take 50 seconds. For each defective screw detected, the operation will be delayed by ½ second.

i) Write the variable Y: the time required to complete the control of quality as a linear transformation of X.

ii) Determine the average time to complete quality control.

Interpret your answer using the law of large numbers.