But I'm slightly lost here. "if the Sidewalk is Slippery then it is probably Wet and this can be explained by either the Sprinkler or the Rain but probably not both, i.e. if we're told that it's Raining we conclude that it's less likely that the Sprinkler was on." This sentence seems... Wrong. If we're told that it's Raining, we conclude that the chances of Sprinkler is... Exactly the same as it was before we learned that the sidewalk was wet.

The probability of Sprinkler goes up when we learn the sidewalk is Slippery, but then down -- but not below its original level -- when we learn that it is raining. (Note that the example is a little counterintuitive, in that it stipulates that Sprinkler and Rain are independent, given Season. In reality, people don't usually turn their sprinklers on when it is raining, a fact which would be represented by an arrow from Rain to Sprinkler. If that connection was added, the probability of Sprinkler would drop close to zero when Rain was observed.)

It's the same with Alarm/Burglar/Earthquake. The probability of Burglar and Earthquake both go up when Alarm is observed. When further observation increases the probability of Burglar, the probability of Earthquake drops, but not below its original level.

In the limiting case where Alarm is certain to be triggered by Burglar or Earthquake but by nothing else, and Burglar and Earthquake have independent probabilities of b and e, then hearing the Alarm raises the probability of Earthquake to e/(b+e-be). The denominator is the probability of either Burglar or Earthquake. Discovering a burglar lowers it back to e.