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Question

Three events $$ A $$, B and $$ C $$ have probabilities $$ \dfrac{2}{5}, \dfrac{1}{3}$$ and $$\dfrac{1}{2} $$ respectively. Given that $$ P(A \cap B)=\dfrac{1}{5} $$ and $$ P(B \cap C)=\dfrac{1}{4}, $$ find the value of $$ P(C \mid B) $$ and $$ P\left(A^{\prime} \cap C^{\prime}\right) $$


Solution

Here, $$ P(A)=\dfrac{2}{5}, P(B)=\dfrac{1}{3}, P(C)=\dfrac{1}{2} \cdot P(A \cap C)=\dfrac{1}{5} $$ and $$ P(B \cap C)=\dfrac{1}{4} $$

$$ \therefore P(C / B)=\dfrac{P(B \cap C)}{P(B)}=\dfrac{1 / 4}{1 / 3}=\dfrac{3}{4} $$

And $$ P\left(A^{\prime} \cap C\right)=1-P(A \cup C)=1-[P(A)+P(C)-P(A \cap C)] $$

$$ =1-\left[\dfrac{2}{5}+\dfrac{1}{2}-\dfrac{1}{5}\right]=1-\left[\dfrac{4+5-2}{10}\right]=1-\dfrac{7}{10}=\dfrac{3}{10} $$

Mathematics

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