2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.

1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
4. Use this Poisson distribution to find the probability that there are
   (A) exactly two genes that mutate,
   (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.