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QUESTION

 There are two railway routes A and B between two towns. On
   route A 30\% of the trains arrive late, while on route B 50\% of the trains arrive late. A
   businessman travels twice as often by route A as he does by
   route B. On a certain day his train is late, What is the
   probability that he traveled by route B that day?

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ANSWER

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\begin{picture}(8,4)

\put(0,2){$\bullet$} \put(0.2,2.1){\line(2,1){2}}
\put(0.2,2.1){\line(2,-1){2}} \put(2.2,3){$A$} \put(2.2,1){$B$}
\put(2.5,3.1){\line(4,1){2}} \put(2.5,1.1){\line(4,1){2}}
\put(4.6,3.5){late} \put(4.6,1.5){late} \put(1,2.8){$\frac{2}{3}$}
\put(1,1.9){$\frac{1}{3}$} \put(3,3.4){$0.3$} \put(3,1.4){$0.5$}

\end{picture}

   Given $P(A)=2P(B) \Rightarrow P(A)=\frac{2}{3},
   P(B)=\frac{1}{3}.$

   $P(\textrm{late}|A)=0.3,
   P(\textrm{late}|B)=0.5$
   P(late)$= \frac{2}{3} \times 0.3+ \frac{1}{3} \times
   0.5=\frac{11}{30}$
   P(B|late)$=\frac{\frac{1}{3} \times
   0.5}{\frac{11}{30}}=\frac{5}{11}$ (by Bayes' formula)

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