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Thread: Two proofs (probability)

  1. #1
    Tau
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    Two proofs (probability)

    Hello

    Help me prove these two classical rules

    1. A and B are two events. Show that the probability that exactly one of these events will occur is $\displaystyle P(A) + P(B) - 2P(A \cap B)$

    2. We know that $\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
    Prove that $\displaystyle P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$

    Thank you.
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    Quote Originally Posted by Tau View Post
    1. A and B are two events. Show that the probability that exactly one of these events will occur is $\displaystyle P(A) + P(B) - 2P(A \cap B)$

    2. We know that $\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
    Prove that $\displaystyle P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$
    Exactly one is: $\displaystyle P(A\cap B^c)+P(B\cap A^c)$.
    Recall that $\displaystyle P(A\cap B^c)=P(A)-P(A\cap B).$

    In #2 let $\displaystyle D=A\cup B$. The expand $\displaystyle P(D\cup C)$.
    The go back and make subsitutions.
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    Hello, Tau!

    2. We know that $\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)$.


    Prove that: .$\displaystyle P(A \cup B \cup C) \; = \;\begin{Bmatrix} P(A) + P(B) + P(C) \\
    - P(A \cap B) - P(A \cap C) - P(B \cap C) \\ + P(A \cap B \cap C) \end{Bmatrix}$

    $\displaystyle P(A \cup B \cup C)$

    . . $\displaystyle =\qquad\qquad\qquad\qquad P\bigg[(A \cup B) \cup C\bigg]$

    . . $\displaystyle =\qquad\qquad P(A \cup B) + P(C) - P\bigg[ (A \cup B) \cap C\bigg] $

    . . $\displaystyle =\qquad\qquad P(A \cup B) + P(C) - \underbrace{P\bigg[(A \cap C) \cup (B \cap C)\bigg]} $

    . . $\displaystyle =\qquad P(A \cup B) + P(C) - \overbrace{\bigg[P(A\cap C) + P(B\cap C) - P([A \cap C] \cap [B \cap C])\bigg]} $


    . . $\displaystyle =\qquad \underbrace{P(A \cup B)} + P(C) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) $

    . . $\displaystyle =\;\overbrace{P(A) + P(B) - P(A \cap B)} + P(C) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$


    . . $\displaystyle =\; P(A) + P(B) + P(C) - P(A\cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$


    . . . . . . . . . . . . ta-DAA!

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