Confidence Intervals · Superchallenge problem

A coin with $P(\text{heads}) = \tfrac12$ is tossed $n = 10$ times, and the usual 95% interval $\hat p \pm 1.96\sqrt{\hat p(1-\hat p)/10}$ is built from the proportion of heads. What is the EXACT probability that this interval contains $\tfrac12$? Give a fraction.