I have got stuck for days regarding a probability calculation.
Given $X|\mu\sim \mathcal{N}(\mu,\sigma^2), \mu \sim \mathcal{N}(\eta, \tau)$ and $C_1, C_2$ are positive reals.
$$ E[E[\mathbf{1}\{X - C_1\sigma < \mu < X + C_1\sigma\} \mathbf{1}\{\eta - C_2\tau < X < \eta + C_2\tau\}|\mu]]$$
I first tried to start from the inner expectation and then next. But it becomes complicated by involving two indicator functions at the same time. I tried to use the posterior then it becomes very messy in the end. It is still possible there would be no closed form.
It would be very appreciated if you could give me any inputs.