There's the easy way and and there's the hard way. Being lazy by nature, I'll outline the key results for the easy way:

If X and Y are jointly gaussian (bivariate normal distribution), then the conditional pdf for one of the variables, given a known value for the other variable, is normally distributed:

$\displaystyle f(x | y = y_0) = \text{Normal} \left(\mu_{x|y = y_0}, \, \sigma^2_{x|y=y_0}\right)$

where:

$\displaystyle \mu_{x|y = y_0} = \mu_x + \rho \sigma_x \frac{(y_0 - \mu_y)}{\sigma_y}$

$\displaystyle \sigma_{x|y=y_0} = \sigma_x \sqrt{1 - \rho^2}$

and $\displaystyle \rho$ is the correlation coefficient.

I'll leave it to you to substitute the necessary notation and values. It should then be routine to calculate $\displaystyle \Pr(C > \mu_C | P = \mu_P)$.

Some reading:

http://www.aos.wisc.edu/~dvimont/aos...iate_notes.pdf