(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that [itex]P(A \cap B)≥1-P(\bar{A})-P(\bar{B})[/itex]

for all [itex] A, B \subseteq S[/itex]

using only these axioms:

1) [itex]0 \leq P(A) \leq 1[/itex], for any event [itex]A \subseteq S[/itex]

2) [itex]P(S) = 1[/itex]

3) [itex] P(A \cup B) = P(A) + P(B)[/itex] if and only if [itex] P(A \cap B) = 0[/itex]

2. Relevant equations

None.

3. The attempt at a solution

My proof:

First rearrange the shizzle:

[itex]P(\bar{A}) + P(A \cap B) + P(\bar{B}) \geq 1[/itex]

Now using the fact that the first two terms are disjoint, use axiom 3 to obtain:

[itex]P(\bar{A} \cup (A \cap B)) + P(\bar{B}) \geq 1[/itex]

Next note that the first term is equals to P(S), hence we get:

[itex]P(S) + P(\bar{B}) \geq 1 [/itex]

which holds for all [itex] B \subseteq S[/itex], because [itex]P(S) = 1[/itex] and [itex] P(\bar{B}) \geq 0 [/itex] for all B.

Is this proof correct?

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# Homework Help: Prove using three basic probability axioms

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