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Vaniver comments on Value of Information: Four Examples - Less Wrong
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Where do you get the exact "half-chance of nothing because you don't play"? How do you decide to play or not, given a favorable outcome of the test run?
You should play if the expected value is positive, and not if it's negative. If the test run results in heads, then the posterior probability is 2/3rds and 24*2/3-12=4, which is positive. If the test run results in tails, then the posterior probability is 1/3rd and 24*1/3-12=-4, which is negative.
(Why is the posterior probability 2/3 or 1/3? Check out footnote 3, or Laplace's Rule of Succession.)