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QUESTION

You are given the following probabilities relating to two
   events A and B, $P(A)=0.5, P(B)=0.7, P(A \mathrm{\ or\ }B)=0.8$. Calculate

    \begin{description}

     \item[(i)]$P(A \textrm{ and } B)$

     \item[(ii)]$P(A \textrm{ and not }B)$

     \item[(iii)]$P(A|B)$

    \end{description}

ANSWER
 \begin{description}

    \item[(i)]
\begin{eqnarray*} P(A \textrm{ and }B)&=&P(A)+P(B)_P(A
      \textrm{ or }B) \textrm{ by addition theorem}\\
      &=&0.5+0.7-0.8=0.4
     \end{eqnarray*}

    \item[(ii)]$P(A \textrm{ and not }B)+ P(A \textrm{ and}
     B)=P(A)$  (since (A and not B) or (A and B)=A, (A and not B)
     and (A and B)=$\phi$ Therefore\\
     $P(A \textrm{ and not }B)=0.5-0.4=0.1$

    \item[(iii)]$P(B|A)=\frac{P(A \textrm{ and
    }B)}{P(A)}=\frac{0.4}{0.5}=0.8$

   \end{description}

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