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 P 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 nT, where Tn is the number of working days. For which n does the expected value of Xn become maximal? Hint: If n = 1, then there is only one holiday, thus XI = 1 . 364 = 364, If there are two workers, then either 2 = 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

Xn will b maximised at n = 182 & 183

This can be easily build into the Excel Sheet using the hint given in the question. Following is the extract of the complete excel sheet.

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 and 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

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