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# Probability Proofs Using Axioms watch

1. I'm stuck on 3 questions, I'll post the questions and my answers. If anyone could look over them and see any errors or how I could proceed it would be much appreciated. Thanks. Jon.

Prove that if A and B are events then

Question

1. P(A intersection B) >= P(A) + P(B) - 1

P(A union B) = P(A) + P(B) - P(A intersection B)
so rearranging
P(A intersection B) + P(A) + P(B) - P(A union B)
substituting back into the original equation
P(A) + P(B) - P(A union B) >= P(A) + P(B) -1
rearranging
P(A union B) <=1

End.

Quetion
2. P(A triangle B) = P(A) + P(B) - 2P(A union B)

I have no idea how to prove this, I drew a Venn diagram and can easily see that it is correct, but am unsure how to prove it by a series of statements.

3rd question

I don't really understand what I have to do in this question

Let S = {1,2,3,...,10}
and for A = {a1,a2,...,ar} C S (where C is underlined meaning A is a subset of S)

define a probability by

P(A) = (a1+a2+...+ar) / (55).

We could equally write this as

P(A) = 1/55 sigma a (with a is an element of A underneath the sigma)

Verify that this satisfies the axoims for probability.

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