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jimrandomh comments on Open Thread, September, 2010-- part 2 - Less Wrong
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These problems only look similar because you are hiding the assumption that (neighbor's lawn wet) = (rain or sprinkler turned on). From this, P(neighbor's lawn wet | Saturday, predicted rain) = P(rain or sprinkler turned on|.5 chance of rain and .5 chance of sprinkler) = 1-P(~rain|...)P(~sprinkler|...) = .75. But there is no valid similar statement for the coin; the analogous disjunction would be (coin heads) = (2008 coin heads or penny heads), in which case treating the clauses of the right hand side as independent means flipping two coins and checking if at least one of them is heads.
I disagree - that makes no difference. Just change "it will rain" to "my neighbor's lawn will be wet", and "my neighbor's sprinkler will turn on" to "my neighbor's lawn will be wet".