5 A computer store has a stock of 10 laptops to lend to customers while their machines are being repaired. On any particular day the number of laptop loans requested follows the distribution given in Table 5.1. \begin{table}[h] \end{table}

1. Give an efficient rule for using two-digit random numbers to simulate the daily number of requests for laptop loans.
2. Use two-digit random numbers from the list below to simulate the number of loans requested on each of ten successive days. Random numbers: \(23,02,57,80,31,72,92,78,04,07\) The number of laptops returned from loan each day is modelled by the distribution given in Table 5.2, independently of the number on loan (which is always at least 5 ). \begin{table}[h]

   Number returned	0	1	2	3
   Probability	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 3 }\)

   \captionsetup{labelformat=empty} \caption{Table 5.2}
   \end{table}
3. Give an efficient rule for using two-digit random numbers to simulate the daily number of laptop returns.
4. Use two-digit random numbers from the list below to simulate the number of returns on each of ten successive days. Random numbers: \(32,98,01,32,14,21,32,71,82,54,47\) At the end of day 0 there are 7 laptops out on loan and 3 in stock. Each day returns are made in the morning and loans go out in the afternoon. If there is no laptop available the customer is disappointed and never gets a loaned laptop.
5. Use your simulated numbers of requests and returns to simulate what happens over the next 10 days. For each day record the day number, the number of laptops in stock at the end of the day, and the number of customers that have to be disappointed.
   [0pt] [3] To try to avoid disappointing customers, if the number of laptops in stock at the end of a day is 2 or fewer, the store sends out e-mails to customers with loaned laptops asking for early return if possible. This changes the return distribution for the next day to that given in Table 5.3. \begin{table}[h]

   Number returned	0	1	2	3	4
   Probability	0.1	0.1	0.4	0.2	0.2

   \captionsetup{labelformat=empty} \caption{Table 5.3}
   \end{table}
6. Simulate the 10 days again, but using this new policy. Use the requests you produced in part (ii). Use the random numbers given in part (iv) to simulate returns, but use either the distribution given in Table 5.2 or that given in Table 5.3, depending on the number of laptops in stock at the end of the previous day. Is the new policy better?