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We introduce a natural way to construct a random subset of a homogeneousCantor set $C$ in $[0,1]$ via random labelings of an infinite binary tree. Fora fixed parameter $p\in(0,1)$, each edge in the tree is independently labeled$0$ with probability $p$, or $1$ with probability $1-p$. Each infinite labeledpath down the tree thus determines a unique point in $C$, and the collective ofall these points is our random subset. When $p=1/2$, we show that the randomsubset is almost surely of full Hausdorff dimension in $C$ but of zeroHausdorff measure. For $p\neq 1/2$ we prove non-trivial upper and lower boundsfor the Hausdorff dimension. We also consider deterministic labelings of thetree where, for a fixed integer $m\geq 2$, every $m$th edge is labeled $1$, andcompute the exact Hausdorff dimension of the resulting subset of $C$.

Random subsets of Cantor sets generated by trees of coin flips