Suppose there are two normal random distribution: $$ a\sim N(0,9) \quad b\sim N(0, 16) $$
Person Alice randomly chooses a distribution from $a$ and $b$ with equal probability (50%) and then draw a number from that distribution.
Alice shows Bob this number and Bob observes the value $2$.
What is the probability that this number is from distribution $a$?
HINT:
$P\{pick \ 2\} = P\{pick \ 2 \ | choose \ a\}\cdot P\{choose \ a\} + P\{pick \ 2 \ | choose \ b\}\cdot P\{choose \ b\} = \frac{1}{2}\cdot f_{a}(2)+ \frac{1}{2}\cdot f_{b}(2)$
where $f_{a}(2)$ is the pdf of $a\sim N(0,9)$ and $f_{b}(2)$ is the pdf of $b\sim N(0, 16)$