$1.$ An important machine is known to never last more than four months. During its first month of operation, it fails $\backslash (10\backslash \%\backslash )$ of the time. If the machine completes its first month, then it fails during its second month $\backslash (20\backslash \%\backslash )$ of the time. If the machine completes its second month of operation, then it will fail during its third month $\backslash (50\backslash \%\backslash )$ of the time. If the machine completes its third month, then it is sure to fail by the end of the fourth month. At the beginning of each month, we must decide whether or not to replace our machine with a new machine. It costs $$500$ to purchase a new machine, but if a machine fails during a month, we incur a cost of $$1,000$