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1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii).
[Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A, i.e. A, n Aj = ø for any i+j, 71 Remark 1.1. 0 E() and S E Ω. For example, a coin us jipped. The sarnple space is S -
Proposition 1.3. The probability function P has the following properties: Ac Ω, P(A)-1-P(A). ()P(0)0 (iii) If A c B, P(A) P(B). urm) Inclusion-Exclusion Principle. VA, B E Ω, P(A U B) P(A) + P(B)-P(A n B). /
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