5. ()A factory employs n workers. The factory is closed at each day of the year

Question

5. ()A factory employs n workers. The factory is closed at each day of the year where at least one of the n workers has his birthday. Only the remaining of the 365 days are working days (days where nobody has birthday). Let Xn be the overall working time, that is, Xn-nTn where Tn is the number of working days. For which n does the expected value of X become maximal? Hint: If n 1, then there is only one holiday, thus X 1 364 364. If there are two workers, then either X2 2-363 if the two workers have their birthdays at different days or X2 2 364 if they have a common birthday. And so on for n 3.4.. .. workers.

Explanation / Answer

This question can be easily answered in the excel sheet by creating the following variables as in the chart below.

Tn = The number of days Open

Xn = n*Tn

Extrapolating this in excel by forming a table with n, Closed Days, Tn & Xn (= n*Tn). You will observe that the value of Xn increases and it maximises at n = 182 and 183. At n=182 & 183 Tn = 33,306 and after that the value of Tn starts decreasing. This is shown in the table below.

n Closed Days Open Days (Tn) Xn = n*Tn 1 1 364 1*364 = 364 2 2 363 2*363 = 726 3 3 362 3*362 = 1086 4 4 361 4*361 = 1444 an so on.. 180 180 185 180*185 = 33,300 181 181 184 181*184 = 33,304 182 182 183 182*183 = 33,306 183 183 182 183*182 = 33,306 184 184 181 184*181 = 33,304 185 185 180 185*180 = 33,300 186 186 179 186*179 = 33,294

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