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Thread: A problem come from "Introduction to Probability Models"

  1. #1
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    A problem come from "Introduction to Probability Models"

    " If the occurrence of B makes A more likely, does the occurrence of A make B more likely?

    I guess yes. But I could not figure out the reason. Thanks~~
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    Hello, shiningstarpxx!

    An interesting question . . .


    If the occurrence of $\displaystyle B$ makes $\displaystyle A$ more likely,
    does the occurrence of $\displaystyle A$ make $\displaystyle B$ more likely?
    The answer is "Yes".
    We need Bayes' Theorem: .$\displaystyle P(X\,|\,Y) \;=\;\frac{P(X \wedge\,Y)}{P(Y)}$


    We are told that, if $\displaystyle B$ happens, $\displaystyle A$ is more likely to happen.
    . . That is, the probability of $\displaystyle A$, given $\displaystyle B$, is greater than the probability of $\displaystyle A$.

    In symbols: .$\displaystyle P(A\,|\,B) \:> \:P(A)$

    This means: .$\displaystyle \frac{P(A \wedge B)}{P(B)} \:> \:P(A) \quad\Rightarrow\quad P(A\wedge B) \:>\:P(A)\!\cdot\!P(B)$

    Hence, we have: .$\displaystyle \frac{P(B \wedge A)}{P(A)} \:>\:P(B) \quad\Rightarrow\quad P(B\,|\,A) \:>\:P(B)$


    Therefore, the probability of $\displaystyle B$, given $\displaystyle A$, is greater than the probability of $\displaystyle B.$
    . . The occurence of $\displaystyle A$ makes $\displaystyle B$ more likely.

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  3. #3
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    Quote Originally Posted by shiningstarpxx View Post
    " If the occurrence of B makes A more likely, does the occurrence of A make B more likely?

    I guess yes. But I could not figure out the reason. Thanks~~
    What you're asking is the following:

    Does Pr(A | B) > Pr(A) => Pr(B | A) > Pr(B) ?

    The answer is yes. Consider:

    $\displaystyle \Pr(A | B) \, \Pr(B) = \Pr(B | A) \, \Pr(A) \Rightarrow \frac{\Pr(A | B)}{\Pr(A)} = \frac{\Pr(B | A)}{\Pr(B)}$.

    But it's given that $\displaystyle \Pr(A | B) > \Pr(A) \Rightarrow \frac{\Pr(A | B)}{\Pr(A)} > 1$.

    Therefore $\displaystyle \frac{\Pr(B | A)}{\Pr(B)} > 1$ and the implication follows.


    Edit: Too fast for me this time, Soroban. But I like to think our replies ...... complement each other lol!
    Last edited by mr fantastic; Jun 2nd 2008 at 05:17 AM. Reason: See edit.
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