Q:

Probability of event A ocurring

I am given this information for calculating the probability for event $A$, $P(A)$:

$P(B) = 0.28$
if $B$ occurs, $P(A) = 0.53$
if $B$ does not occur, $P(A) = 0.2$

As someone recently returning to statistics, how would I go about finding $P(A)$?

A:

We can write $ P(A) = P(A \cap B) + P(A\cap B^c)$. 
Now $P(A \cap B) = P(B)P(A|B)$, where $P(A|B)$ denotes the event that $A$ occurs conditioned on the event that $B$ has occured.
So we have $P(A) =  P(B)P(A|B) + P(B^c)P(A|B^c)$, where $B^c$ denotes the event that $B$ does not occur. 
Putting in the values we get
$P(A) = 0.28*0.53 + (1-0.28)*0.2 = 0.2924$.  

