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Let ${E_1},{E_2},{E_3}$ be three arbitrary events of a sample space $S$. Consider the following statements which of the following statements are correct
$P$ (only one of them occurs)
$ = P({\bar E_1}{E_2}{E_3} + {E_1}{\bar E_2}{E_3} + {E_1}{E_2}{\overline E _3})$
$P$ (none of them occurs)
$ = P({\overline E _1} + {\overline E _2} + {\overline E _3})$
$P$ (atleast one of them occurs)
$ = P({E_1} + {E_2} + {E_3})$
$P$ (all the three occurs)$ = P({E_1} + {E_2} + {E_3})$
where $P({E_1})$denotes the probability of ${E_1}$ and ${\bar E_1}$ denotes complement of ${E_1}$.
Solution
(c) $P$ (only one of them occurs)
$ = P({E_1}{\bar E_2}{\bar E_3} + {\bar E_1}{E_2}{\bar E_3} + {\bar E_1}{\bar E_2}{E_3})$
$ \ne P({\bar E_1}{E_2}{E_3} + {E_1}{\bar E_2}{E_3} + {E_1}{E_2}{\bar E_3})$
$\therefore$ $(a)$ is incorrect.
$P$ (none of them occurs)
$ = P({\bar E_1} \cap {\bar E_2} \cap {\bar E_3}) \ne P({\bar E_1} + {\bar E_2} + {\bar E_3})$
$\therefore$ $(b)$ is not correct.
$P$ (atleast one of them occurs)
$ = P({E_1} \cup {E_2} \cup {E_3}) = P({E_1} + {E_2} + {E_3})$
$\therefore$ $ (c) $ is correct.
$P$ (all the three occurs)
$ = P({E_1} \cap {E_2} \cap {E_3}) \ne P({E_1} + {E_2} + {E_3})$
$\therefore$ $(d)$ is not correct.
