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Evaluate $ P(A \cup B)$, if $ 2P(A) = P(B) = \large \frac{5}{13} $ and $P(A | B) = \large \frac{2}{5} $

$\begin{array}{1 1} \frac{11}{26} \\\frac{9}{26} \\ \frac{13}{26} \\ \frac{7}{26} \end{array} $

1 Answer

Toolbox:
  • \(1)\;p(A\;\cap\;B)\;=\;p(A/B)\;p(B)\)
  • \(2)\;p(A\;\cup\;B)\;=\;p(A)\;+\;p(B)\;-\;p(A\;\cap\;B)\)
Given \(2\;p(A)\;=\;p(B)=\Large\frac{5}{13}\)
\(\Rightarrow\;p(A)=\Large\frac{5}{26}\)
\(p(A/B)=\Large\frac{2}{5}\)
$\Rightarrow$ \(p(A\;\cap\;B)=\large\frac{2}{5}\;\times\;\frac{5}{13}\;=\;\frac{2}{13}\)
Since \(\;p(A\;\cup\;B)\;=\;p(A)\;+\;p(B)\;-\;p(A\;\cap\;B)\)
\(p(A\;\cup\;B)=\;\large\frac{5}{26}\;+\;\frac{5}{13}\;-\;\frac{2}{13}\;=\;\frac{11}{26}\)
answered Mar 20, 2013 by poojasapani_1
edited Jun 18, 2013 by balaji.thirumalai
 

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