Mixtures of Independent Distributions Destroy Independence

A biased coin lands heads with probability $1/2$. If heads, set $(X, Y) = (1, 1)$. If tails, draw $X$ and $Y$ independently, each $\text{Bernoulli}(1/2)$. (a) Compute the full joint distribution of $(X, Y)$. (b) Show that $X$ and $Y$ have the same marginal distribution. (c) Are $X$ and $Y$ independent? Prove your answer. (d) Compute $P(X = 1 \mid Y = 0)$ and compare it to $P(X = 1)$. Interpret the result.