When testing insecticides, the amount of the chemical when given all at once will result in the death of 50% of the population is called the LD50, where LD stands for lethal dose. If 40 insects are placed in separate petri dishes and treated with an insecticide dosage of LD50, find the probabilities of the following events: (a): exactly 20 survive (b): find the number expected to survive out of 40 (c): find the variance and standard deviation of the number of survivors out of 40.