Write a function in Python, without using any special packages or libraries, that performs the following:

Calculate P$($A$|$B$)$ given P$($A$),$ P$($B$|$A$),$ and P$($B$|$~A$).$

Parameters:

p$\_$a $($float$)$: Probability of A happening.

p$\_$b$\_$given$\_$a $($float$)$: Probability of B given A$.$

p$\_$b$\_$given$\_$not$\_$a $($float$)$: Probability of B given not A$.$

Returns: Probability of A given B $($float$).$

Test your formula by providing the following values for your parameters:

p$\_$a $($float$)$: $0\mathrm{.}20$

p$\_$b$\_$given$\_$a $($float$)$: $0\mathrm{.}30$

p$\_$b$\_$given$\_$not$\_$a $($float$)$: $0\mathrm{.}10$