Assignment:
A total of M cells are exposed to X-ray radiation. X-rays produce chromosome breakages in cells. For each individual cell i € {1, .. . , M}, the number N2 of breakages has a Poisson distribution with parameter A (the random variables N are independent). Each breakage has a probability q of healing perfectly, a probability p of healing with a mutation, and a probability r = 1 - p - q of not healing (the associated events are independent between breakages). If all the breakages of a cell are not healed perfectly or healed with a mutation then the cell dies.
a) What is the probability that a given cell dies from the radiation?
b) What is the probability that a given cell survives the radiation without any mutations?
c) What is the probability that a given cell survives the radiation with at least one mutation?
d) What is the average number of mutations per surviving cells?
Provide complete and step by step solution for the question and show calculations and use formulas.