Expected Value of the Geometric PMF at Its Own Random Variable

Let $X \sim \text{Geom}(p)$ with PMF $p(k) = P(X = k) = p(1-p)^{k-1}$ for $k = 1, 2, 3, \ldots$ Compute $E[p(X)]$ -- the expected value of the PMF evaluated at the random variable itself -- when $p = 1/3$. **Note:** There is a tempting but incorrect shortcut. Identify the error in the following reasoning, then compute the correct answer: > "$E[p(X)] = E[P(X = X)] = E[1] = 1$."