1. The probability that a system recovers from a rare cold spell is 0.35. If 80 systems are known to be affected by cold spell is a certain year, what is the probability that less than 40 recover?
2. Historical records of sewer pipe failures in an old city show that during the period 2005 to 2013, there were 30 failures that were catastrophic (exploded due to the build-up of methane or other gases). Assuming that the pipe failures follow a Poisson process, find the probability that
(a) 4 catastrophic failures occur within the next 6 months,
(b) the interval from now till the next catastrophic pipe failure will be at most 2 months
(c) The probability that no catastrophic pipe failure will occur in the next 12 months.