If P(A) = 5 , P(B) = 10 and P(A B) = 5 , then P ( A / B ) P ( B / A ) is equal to

Here, P(A) = 5 ,P(B) = 10 and P(A B) = 5 P ( A / B ) = ) P ( B ) = 1 - P(B) = 1 - P(B) = 5 + 10 - 5 ) 1 - 10 = 10 ) 10 = 2 10 = 7 and P ( B / A ) = ) P ( A ) = 1 - P(A) = 2 1 - 5 = 3/5 = 6 P ( A / B ) P ( B / A ) = 7 6 = 42

Probability — Class 12 Maths Solution

exemplar objective MCQ NCERT,Exemp,Q.59,Page.279

Question

If $P(A) = \frac{2}{5}$, $P(B) = \frac{3}{{10}}$ and $P(A \cap B) = \frac{1}{5}$, then $P\left( {{A^\prime }/{B^\prime }} \right) \cdot P\left( {{B^\prime }/{A^\prime }} \right)$ is equal to

(a) $\frac{5}{6}$
(b) $\frac{5}{7}$
(c) $\frac{{25}}{{42}}$ ✓ Correct
(d) 1

Step-by-step Solution

Here, $P(A) = \frac{2}{5},P(B) = \frac{3}{{10}}$ and $P(A \cap B) = \frac{1}{5}$

$P\left( {{A^\prime }/{B^\prime }} \right) = \frac{{P\left( {{A^\prime } \cap {B^\prime }} \right)}}{{P\left( {{B^\prime }} \right)}} = \frac{{1 - P(A \cup B)}}{{1 - P(B)}}$

$= \frac{{1 - [P(A) + P(B) - P(A \cap B)]}}{{1 - P(B)}}$
$= \frac{{1 - \left( {\frac{2}{5} + \frac{3}{{10}} - \frac{1}{5}} \right)}}{{1 - \frac{3}{{10}}}}$

$= \frac{{1 - \left( {\frac{{4 + 3 - 2}}{{10}}} \right)}}{{\frac{7}{{10}}}} = \frac{{1 - \frac{1}{2}}}{{\frac{7}{{10}}}} = \frac{5}{7}$

and $P\left( {{B^\prime }/{A^\prime }} \right) = \frac{{P\left( {{B^\prime } \cap {A^\prime }} \right)}}{{P\left( {{A^\prime }} \right)}} = \frac{{1 - P(A \cup B)}}{{1 - P(A)}}$

$= \frac{{1 - \frac{1}{2}}}{{1 - \frac{2}{5}}} = \frac{{1/2}}{{3/5}} = \frac{5}{6}$

$\therefore$ $P\left( {{A^\prime }/{B^\prime }} \right) \cdot P\left( {{B^\prime }/{A^\prime }} \right) = \frac{5}{7} \cdot \frac{5}{6} = \frac{{25}}{{42}}$

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