Use and
Problem:
Let A and B be two events, show that:
P(A) + P(B) -1 P(A U B) P(A)+P(B)
Questions/Attempts:
Am I supposed to assume that A and B are the only events in the sample space, and that therefore P(A)+P(B)=1?
If I assume that, and begin the proof with the first axiom of probability, I get this far:
1. 0 P(A) 1
2. P(A)+P(B)=1 (Since P(S)=1, where P(S) is the sample space)
so
3. 0 P(A) P(A) + P(B)
There's a couple things I can do from here but it seems to make things more complicated than they need to be.
Thanks for any help in advance!
Questions:
1. If I reduce the problem statement to an identity, is that considered "proving" it?
2. In this problem, can I assume P(A)+P(B)=1?
Attempt:
1.P(A) + P(B) -1 P(A U B) P(A)+P(B)
2. P(A) + P(B) -1 P(A)+P(B)- P(A B ) P(A)+P(B)
but P(A)+P(B)=1 so,
3. 0 1 - P(A B ) 1
and P(A B ) is between 0 and 1.
It almost seems like the proof went backwards here, from a statement to an identity. Is this correct? Thanks for your help Plato I appreciate it.