Probability Question 390
Question: 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
[Pb. CET 2004]
Options:
A) P (only one of them occurs) $ =P({{\bar{E}}_1}E_2E_3+E_1{{\bar{E}}_2}E_3+E_1E_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 $ .
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Answer:
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_2E_3+E_1{{\bar{E}}_2}E_3+E_1E_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_2E_3+E_1{{\bar{E}}_2}E_3+E_1E_2{{\bar{E}}_3}) $ is incorrect.
BETA
