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Math Help - Question on Probability

  1. #1
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    Question on Probability

    Given that
    P(\neg A)=0.6
    P(B\mid A)=0.7
    P(B)=0.3
    What is P(\neg A \wedge B)?
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  2. #2
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    Quote Originally Posted by quiney View Post
    Given that
    P(\neg A)=0.6
    P(B\mid A)=0.7
    P(B)=0.3
    What is P(\neg A \wedge B)?
    A simple approach would be to draw a tree diagram. Have you tried doing that?
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  3. #3
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    Here is another way.
    Recall that P(B) = P(B \cap A) + P(B \cap \neg A)
    Solve for  P(B \cap \neg A) .
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  4. #4
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    Hello, quiney!

    Yet another approach . . .


    Given that:
    . . \begin{array}{ccc}P(\sim\!A)&=& 0.6 \\<br />
P(B\,|\,A) &=& 0.7 \\<br />
P(B) &=& 0.3 \end{array}

    What is . P(\sim\!A \wedge B) ?

    We can place the data into a chart:

    . . \begin{array}{c||c|c||c}<br />
& B & \sim\!B & \text{total} \\ \hline \hline<br />
A & & & 0.40 \\ \hline<br />
\sim\!A & & & 0.60 \\ \hline \hline<br />
\text{Total} & 0.30 & 0.70 & 1.00 \end{array}


    We have: . P(B|A) \:=\:0.7 \quad\Rightarrow\quad \dfrac{P(B \wedge A)}{P(A)} \:=\:0.7

    . . . . P(B \wedge A) \:=\:0.7\cdot P(A) \;=\;(0.7)(0.4)

    . . . . P(A \wedge B) \;=\;0.28


    Insert that into the chart.
    . . Fill in the rest of the chart.

    . . \begin{array}{c||c|c||c}<br />
& B & \sim\!B & \text{total} \\ \hline \hline<br />
A & 0.28 & 0.12 & 0.40 \\ \hline<br />
\sim\!A & 0.02 & 0.58 & 0.60 \\ \hline \hline<br />
\text{Total} & 0.30 & 0.70 & 1.00 \end{array}


    Therefore: . P(\sim\!A \wedge B) \;=\;0.02



    Corrected my typo . . . Thanks, Plato!
    Last edited by Soroban; August 17th 2010 at 04:30 PM.
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  5. #5
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    Quote Originally Posted by Soroban View Post
    We have: . P(B|A) \:=\:0.7 \quad\Rightarrow\quad \dfrac{P(A \wedge B)}{P(B)} \:=\:0.7
    Please note the typo  \displaystyle P(B|A)=\frac{P(A\cap B)}{P(A)}.
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  6. #6
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    Thanks a lot everybody. So, since p(B)=p(B \wedge A) + p(B \wedge \neg A):
    0.3=p(B \wedge A) + p(B \wedge \neg A)

    Since p(B|A)=0.7=\dfrac{p(B \wedge A)}{p(A)=0.4} \quad\Rightarrow\quad\ 0.28=p(B\wedge A) \quad\Rightarrow\ 0.3=0.28 + p(B \wedge\neg A)
    Therefore
    0.02=p(B \wedge\neg A)=p(\neg A \wedge B)

    Funny, the answer guide in the book says 0.2. Must be a typo.
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