The amounts of rice, by weight, in 20 nominally 500g packets are determined. The weights, in $\mathrm{g}$ , are as follows.  

496 506 495 491 488 492 482 495 493 496   
487 490 493 495 492 498 491 493 495 489  

Assume that, given the values of parameters $\mu,\ \tau$ , the weights are independent and each has a normal $N(\mu,\,\tau)$ distribution.  

The values of $\mu$ and $\tau$ are unknown. Our prior distribution is as follows. We have a gamma(2, 9) prior distribution for $\tau$ and a $N(500,\ (0.005\tau)^{-1})$ conditional prior distribution for $\mu$ given $\tau$ .  

(a) Find the posterior probability that $\mu<495$ .  

(b) Find the posterior predictive probability that a new packet of rice will contain less than 500g of rice  