2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

1. Explain the meaning of the term 'independently' and name the distribution that models this situation.
2. Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
3. Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
4. A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
5. State the exact distribution of the number of boxes which have at least 8 defects.
6. Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.