No empty box after a geometric number of balls

A fair coin is tossed until the first head; if that happens on toss $k$, you receive $k$ balls. Each ball is then dropped into one of three labelled boxes, chosen uniformly and independently. What is the probability that every box ends up with at least one ball?

$\tfrac1{10}$. Given $k$ balls, inclusion-exclusion gives $P(\text{no empty box})=1-3(\tfrac23)^k+3(\tfrac13)^k$. Averaging against $P(k)=2^{-k}$ uses geometric sums: $1-3\cdot\tfrac{1/3}{2/3}+3\cdot\tfrac{1/6}{5/6}=1-\tfrac32+\tfrac35=\tfrac1{10}$.