$4.$B$.2$ Processing the test results of a laboratory experiment requires a lot of computer calculations. It is therefore decided to perform the calculations at night. The experiment consists of $100$ tests and thus results in $100$ test results. Test processing times are independent and exponentially distributed with a mean of $10$ minutes. The computer processes the test results one by one and starts calculating at $6pm.$

a$.$ What is the probability that that the processing of the test results is finished before the start of the next workday, at $9$ am $($i$.$e$.,$ within $900$ minutes$)?$ Make both an exact and an approximate calculation $($the latter with the normal approximation$).$

b$.$ If it turns out that the computer is still working the next morning $(9am),$ more precisely with the ${89}^{th}$ test result, what is then the expectation of the remaining processing time?