Indepedence mutually exclusiveness

#1
I'm working my way through my ACTEX study guide for the probability exam, and a question is really tripping me up. It goes like this:

You are given that p(a) = 0.5 and p(A union B) = 0.7.

Actuary 1 assumes A and B are independent and calculates p(B) based on that assumption.

Actuary 2 assumes A and B are mutually exclusive and calculates p(B) based on that assumption.


Calculating based on mutually exclusiveness isn't what's giving me problems, it's calculating when assuming independence.

If A and B are independent, and it says that the probability of A and B occuring equals 0.7, why isn't this problem solved by setting up an equation like this:

0.5 p(b) = 0.7 and solving for B? In my study guide, they solve it by using the compliments. Why is this only solved using the compliments? I don't understand the logic behind it. I know that it must be that way because if i solved it my way, the probability of B would be over 1. What am I missing here?

Furthermore, if i plugged in the probability of B, which is equal to 0.4 and multiply it by A, which is 0.5, it doesn't equal 0.7... I am so confused. XD