How many printers get at least one job?

Question

An office has 4 printers. Three print jobs are sent out, and each job independently goes to one of the 4 printers chosen uniformly at random.

On average, how many printers end up handling at least one job?

Accepted Answer

Counting busy printers directly means juggling all the ways 3 jobs can spread over 4 machines. Instead, put one indicator on each printer and add. Fix a printer. It stays idle only if all 3 jobs avoid it. Each job misses a given printer with probability 	frac34, so P(	ext{a given printer is idle}) = \left(	frac34ight)^3 = \frac{27}{64}. Then the chance that printer is busy (has at least one job) is 1 - 	frac{27}{64} = 	frac{37}{64}. Let Xi = 1 if printer i is busy; E[Xi] = 	frac{37}{64}. With 4 printers, X = X1 + \cdots + X4, so by linearity E[X] = 4 	imes \frac{37}{64} = \frac{37}{1…

How many printers get at least one job?

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