For a continuous $X \sim \mathcal{N}(\mu, \sigma^2)$, return the tuple (p_exact, p_at_least):

• p_exact is $P(X = x)$
• p_at_least is $P(X \geq x)$

The first one catches people out. A continuous distribution assigns probability to intervals, not to points, so the probability of any single exact value is zero; the density at x is not a probability.

Use the error function for the tail: $P(X \geq x) = \tfrac{1}{2}\,\mathrm{erfc}\!\left(\tfrac{x-\mu}{\sigma\sqrt{2}}\right)$

Input

mu = 0.0
sigma = 1.0
x = 0.0

Output

(0.0, 0.5)