If you roll a single die, what is the expected number of rolls necessary to roll every number once?
Think of it like six mini-games. For each game, we roll the die until we roll a number that hasn't been rolled yet—what we'll call a "win". Then we start the next game.
The expected value of each Geometric random variable is
For the first game,
For the second game, 5 out of the 6 outcomes are new, so
For the third game, 4 of the 6 possible rolls are new, so
By this point, we can see a pattern. Since the number of "winning" rolls decreases by 1 for each new game, the probability of "winning" each game goes down from
#"E"(X) = "E"(X_1+X_2+X_3+X_4+X_5+X_6)#
#color(white)("E"(X)) = "E"(X_1)+"E"(X_2)+...+"E"(X_5)+"E"(X_6)#
#color(white)("E"(X)) = 6/6+6/5+6/4+6/3+6/2+6/1#
#color(white)("E"(X)) = 1+1.2+1.5+2+3+6#
#color(white)("E"(X)) = 14.7#