Proving identities using the axioms of probability

[phosgene]

Homework Statement

If A and B are events, use the axioms of probability to show:

a) If [itex]A \subset B[/itex], then [itex]P(B \cap A^{C}) = P(B) - P(A)[/itex]

b) [itex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/itex]

Homework Equations

Axiom 1: [itex]P(x)\geq 0[/itex]

Axiom 2: [itex]P(S) = 1[/itex], where S is the state space.

Axiom 3: If [itex]A_{1},A_{2},...,A_{n},...[/itex] is any set of disjoint events, then:

[itex]P(\bigcup_{i} A_{i})=\sum_{i} P(A_{i}) [/itex]

The Attempt at a Solution

It's easy to see why they are true using venn diagrams. In the first case, since A is a subset of B, the probability of the intersection of B and A complement is just the probability of B minus the probability of A. With the second one, you just want to add the P(A) and P(B) and then subtract the probability of the intersection so that you're not adding the probability of those events twice.

I just have no idea how to do this symbolically. This is the first time I've ever had to try and prove something in probability..

 

[Ray Vickson]

Homework Statement

If A and B are events, use the axioms of probability to show:

a) If [itex]A \subset B[/itex], then [itex]P(B \cap A^{C}) = P(B) - P(A)[itex]

b) [itex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/itex]

Homework Equations

Axiom 1: [itex]P(x)\geq 0[/itex]

Axiom 2: [itex]P(S) = 1[/itex], where S is the state space.

Axiom 3: If [itex]A_{1},A_{2},...,A_{n},...[/itex] is any set of disjoint events, then:

[itex]P(\bigcup_{i} A_{i})=\sum_{i} P(A_{i}) [/itex]

The Attempt at a Solution

It's easy to see why they are true using venn diagrams. In the first case, since A is a subset of B, the probability of the intersection of B and A complement is just the probability of B minus the probability of A. With the second one, you just want to add the P(A) and P(B) and then subtract the probability of the intersection so that you're not adding the probability of those events twice.

I just have no idea how to do this symbolically. This is the first time I've ever had to try and prove something in probability..

For (a): ##B = (B \cap A^c) \cup \{\text{something else disjoint}\}##. What must be that "something else"? What do the basic probability axioms then give you?

 

[phosgene]

I use axiom three and re-arrange the result to get the statement that I was trying to prove. Thanks :)