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Consider the Bayesian network in Figure burglary-figure.

  1. If no evidence is observed, are ${Burglary}$ and ${Earthquake}$ independent? Prove this from the numerical semantics and from the topological semantics.

  2. If we observe ${Alarm}{{,=,}}{true}$, are ${Burglary}$ and ${Earthquake}$ independent? Justify your answer by calculating whether the probabilities involved satisfy the definition of conditional independence.