Let E and F be two independent events. The pr | vedclass

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Question

(a) We are given $P(E \cap F) = \frac{1}{{12}}$ and $P(\bar E \cap \bar F) = \frac{1}{2}$
$ \Rightarrow P(E)\,.\,P(F) = \frac{1}{{12}}$.....$(i)$
an

Vedclass · Accepted Answer

$P\,(E) = \frac{1}{3},\,\,P\,(F) = \frac{1}{4}$

Vedclass · Answer

(a) We are given $P(E \cap F) = \frac{1}{{12}}$ and $P(\bar E \cap \bar F) = \frac{1}{2}$
$ \Rightarrow P(E)\,.\,P(F) = \frac{1}{{12}}$.....$(i)$
and $P(\overline E )\,.\,P(\bar F) = \frac{1}{2}$.....$(ii)$
$ \Rightarrow \{ (1 - P(E)\} $ $\{ (1 - P)(F)\} = \frac{1}{2}$
$ \Rightarrow 1 + P(E)P(F) - P(E) - P(F) = \frac{1}{2}$
$ \Rightarrow 1 + \frac{1}{{12}} - [P(E) + P(F)] = \frac{1}{2}$
$ \Rightarrow P(E) + P(F) = \frac{7}{{12}}$.....$(iii)$
On solving $(i)$ and $(iii),$ we get
$P(E) = \frac{1}{3},\,\frac{1}{4}$ and $P(F) = \frac{1}{4},\,\frac{1}{3}$.

Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happens is $\frac{1}{{12}}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2},$ then

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