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.
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.)