Determine whether each of the following is true or false (note: the statement is true if it is always true, otherwise it is false). If you s

Question

Determine whether each of the following is true or false (note: the statement is true if it is always true, otherwise it is false). If you say it is true than give a proof, while if you say it is false then give a counterexample, i.e., a particular case where it fails.

(a) A − B = A ∪ B

(b) Pr (A ∪ B) ≤ Pr (A) + Pr (B)

(c) Pr (A|B) ≥ Pr (A) if B ⊃ A

(d) If A1 . . . , An are mutually exclusive, then Pr (B) = Pn i=1 Pr (B|Ai) Pr (Ai)

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Camila 3 months 2022-02-19T06:27:09+00:00 1 Answer 0 views 0

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    2022-02-19T06:28:33+00:00

    Answer:

    Step-by-step explanation:

    We are given four statements and we must check whether true or false

    a) A − B = A ∪ B

    False

    Eg: A={1,2} B = {2,3}

    A-B = {1}

    AUB ={1,2,3} not equal

    (b) Pr (A ∪ B) ≤ Pr (A) + Pr (B)

    True because

    Pr (A ∪ B)=Pr (A) + Pr (B)
    -P(AB), where P(AB)≥0

    (c) Pr (A|B) ≥ Pr (A) if B ⊃ A

    If A is a subset of B, we have P(AB) = P(A)

    So P(A/B ) = P(A)/P(B) ≥P(A) since P(B) is always less than or equal to 1

    (d) If A1 . . . , An are mutually exclusive, then Pr (B) = Pn i=1 Pr (B|Ai) Pr (Ai)

    P(B) = [tex]\Sigma P(A_i B)[/tex]

    =[tex]\Sigma _1^n P(A_i) P(B/A_i)[/tex]

    so true

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45:7+7-4:2-5:5*4+35:2 =? ( )