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PostPosted: Wed, 15 Sep 2010 21:39:53 UTC 
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Q:If C1,C2,C3,...,Ck are k events in he sample space C,show that the probability that at least one of the events occurs is one minus the probability that none of them occur;i.e.,
P(C1UC2UC3...UCk) = 1- P(C'1& C'2 & C'3 ... & C'k).


where U= union( of events)
& = intersection (of events)
' = means complement


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PostPosted: Thu, 16 Sep 2010 04:16:08 UTC 
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amjadsuhail wrote:
Q:If C1,C2,C3,...,Ck are k events in he sample space C,show that the probability that at least one of the events occurs is one minus the probability that none of them occur;i.e.,
P(C1UC2UC3...UCk) = 1- P(C'1& C'2 & C'3 ... & C'k).


where U= union( of events)
& = intersection (of events)
' = means complement


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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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PostPosted: Thu, 16 Sep 2010 08:34:09 UTC 
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amjadsuhail wrote:
Q:If C1,C2,C3,...,Ck are k events in he sample space C,show that the probability that at least one of the events occurs is one minus the probability that none of them occur;i.e.,
P(C1UC2UC3...UCk) = 1- P(C'1& C'2 & C'3 ... & C'k).

This boils down to set theory, friend:

$(C_1\cup C_2\cup\dotsb \cup C_k)'=C_1'\cap C_2'\cap\dotsb\cap C_k'


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