A) P (only one of them occurs) \[=P({{\bar{E}}_{1}}{{E}_{2}}{{E}_{3}}+{{E}_{1}}{{\bar{E}}_{2}}{{E}_{3}}+{{E}_{1}}{{E}_{2}}{{\overline{E}}_{3}})\]
B) P (none of them occurs) \[=P({{\overline{E}}_{1}}+{{\overline{E}}_{2}}+{{\overline{E}}_{3}})\]
C) P (at least one of them occurs) \[=P({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\]
D) 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 \[{{E}_{1}}\].
Correct Answer: C
Solution :
P (only one of them occurs) \[=P({{E}_{1}}{{\bar{E}}_{2}}{{\bar{E}}_{3}}+{{\bar{E}}_{1}}{{E}_{2}}{{\bar{E}}_{3}}+{{\bar{E}}_{1}}{{\bar{E}}_{2}}{{E}_{3}})\] \[\ne P({{\bar{E}}_{1}}{{E}_{2}}{{E}_{3}}+{{E}_{1}}{{\bar{E}}_{2}}{{E}_{3}}+{{E}_{1}}{{E}_{2}}{{\bar{E}}_{3}})\] \ is incorrect. P (none of them occurs) \[=P({{\bar{E}}_{1}}\cap {{\bar{E}}_{2}}\cap {{\bar{E}}_{3}})\ne P({{\bar{E}}_{1}}+{{\bar{E}}_{2}}+{{\bar{E}}_{3}})\] \ is not correct. P (atleast one of them occurs) \[=P({{E}_{1}}\cup {{E}_{2}}\cup {{E}_{3}})=P({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\] \ is correct. P (all the three occurs) \[=P({{E}_{1}}\cap {{E}_{2}}\cap {{E}_{3}})\ne P({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\] \ is not correct.P (only one of them occurs) \[=P({{E}_{1}}{{\bar{E}}_{2}}{{\bar{E}}_{3}}+{{\bar{E}}_{1}}{{E}_{2}}{{\bar{E}}_{3}}+{{\bar{E}}_{1}}{{\bar{E}}_{2}}{{E}_{3}})\] \[\ne P({{\bar{E}}_{1}}{{E}_{2}}{{E}_{3}}+{{E}_{1}}{{\bar{E}}_{2}}{{E}_{3}}+{{E}_{1}}{{E}_{2}}{{\bar{E}}_{3}})\] \ is incorrect.
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