Pr 1 3 123 And Pr 1 2 Pr 1 3 Pr 2 3 1 9 Pairwise independent Pr 1 2 3 Pr 1 9 Pr

Pr 1 3 123 and pr 1 2 pr 1 3 pr 2 3 1 9 pairwise

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𝑠𝑠𝑠𝑠𝑠𝑠𝑜𝑜𝑁𝑁 𝑖𝑖𝑖𝑖 𝑜𝑜𝑡𝑁𝑁 𝑜𝑜𝑁𝑁𝑖𝑖𝑠𝑠𝑠𝑠𝑁𝑁 𝑖𝑖𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑁𝑁𝑠𝑠𝑖𝑖𝑁𝑁𝐻𝐻 𝑁𝑁𝑏𝑏 𝑠𝑠 } Pr 𝐴𝐴 𝐸𝐸 = 1 3 𝑖𝑖 = 1,2,3 And Pr( 𝐴𝐴 1 ∩ 𝐴𝐴 2 ) = Pr( 𝐴𝐴 1 ∩ 𝐴𝐴 3 ) = Pr ( 𝐴𝐴 2 ∩ 𝐴𝐴 3 ) = 1 9 Pairwise independent Pr( 𝐴𝐴 1 ∩ 𝐴𝐴 2 ∩ 𝐴𝐴 3 ) = Pr({ 𝑠𝑠𝑠𝑠𝑠𝑠 }) = 1 9 Pr( 𝐴𝐴 1 )Pr( 𝐴𝐴 2 )Pr( 𝐴𝐴 3 )
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26 Mutually(Simultaneous) Independence and Pairwise Independence A collection if events 𝐴𝐴 1 , 𝐴𝐴 2 , ⋯ 𝐴𝐴 𝑛𝑛 are mutually independence if for ant subcollection 𝐴𝐴 𝐸𝐸 1 , 𝐴𝐴 𝐸𝐸 2 , ⋯ 𝐴𝐴 𝐸𝐸 𝑘𝑘 ,we have Pr(A i 1 A i 2 ∩ ⋯ ∩ A i k ) = Pr 𝑗𝑗=1 𝑘𝑘 𝐴𝐴 𝐸𝐸 𝑗𝑗 = 𝑗𝑗=1 𝑘𝑘 Pr( 𝐴𝐴 𝐸𝐸 𝑗𝑗 ) = Pr 𝐴𝐴 𝐸𝐸 1 Pr 𝐴𝐴 𝐸𝐸 2 Pr( 𝐴𝐴 𝐸𝐸 𝑘𝑘 ) Example: Consider of tossing a coin three times, the sample space has eight points, namely: { 𝐻𝐻𝐻𝐻𝐻𝐻 , 𝐻𝐻𝐻𝐻𝑇𝑇 , 𝐻𝐻𝑇𝑇𝐻𝐻 , 𝐻𝐻𝑇𝑇𝑇𝑇 , 𝑇𝑇𝐻𝐻𝐻𝐻 , 𝑇𝑇𝐻𝐻𝑇𝑇 , 𝑇𝑇𝑇𝑇𝐻𝐻 , ( 𝑇𝑇𝑇𝑇𝑇𝑇 )} Let 𝐻𝐻 𝐸𝐸 , 𝑖𝑖 = 1,2,3 , denote the event that the 𝑖𝑖 𝑡𝑡𝑡 toss is a 𝐻𝐻 , for example, 𝐻𝐻 1 = {( 𝐻𝐻𝐻𝐻𝐻𝐻 ), ( 𝐻𝐻𝐻𝐻𝑇𝑇 ), ( 𝐻𝐻𝑇𝑇𝐻𝐻 ), ( 𝐻𝐻𝑇𝑇𝑇𝑇 )} Pr 𝐻𝐻 1 ∩ 𝐻𝐻 2 = Pr 𝐻𝐻𝐻𝐻𝐻𝐻 , 𝐻𝐻𝐻𝐻𝑇𝑇 = 2 8 = 1 2 1 2 = Pr 𝐻𝐻 1 Pr( 𝐻𝐻 2 ) Pr 𝐻𝐻 1 ∩ 𝐻𝐻 3 = Pr 𝐻𝐻𝐻𝐻𝐻𝐻 , 𝐻𝐻𝑇𝑇𝐻𝐻 = 2 8 = 1 2 1 2 = Pr 𝐻𝐻 1 Pr( 𝐻𝐻 3 ) Pr 𝐻𝐻 2 ∩ 𝐻𝐻 3 = Pr 𝑇𝑇𝐻𝐻𝐻𝐻 , 𝐻𝐻𝐻𝐻𝐻𝐻 = 2 8 = 1 2 1 2 = Pr 𝐻𝐻 2 Pr( 𝐻𝐻 3 ) Pr 𝐻𝐻 1 ∩ 𝐻𝐻 2 ∩ 𝐻𝐻 3 = Pr 𝐻𝐻𝐻𝐻𝐻𝐻 = 1 8 = Pr 𝐻𝐻 1 Pr 𝐻𝐻 2 Pr( 𝐻𝐻 3 )
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27 Mutually(Simultaneous) Independence and Pairwise Independence Example: Flip a fair coin 4 times. Let 𝐴𝐴 𝐸𝐸 = 𝑠𝑠 𝑡𝑁𝑁𝑠𝑠𝐻𝐻 𝑖𝑖𝑜𝑜 𝑜𝑜𝑁𝑁𝑜𝑜𝑁𝑁𝑁𝑁𝑒𝑒𝑁𝑁𝐻𝐻 𝑜𝑜𝑁𝑁𝑜𝑜𝑁𝑁 𝑜𝑜𝑡𝑁𝑁 𝑖𝑖 𝑡𝑡𝑡 𝑜𝑜𝑠𝑠𝑖𝑖𝑠𝑠 , 𝑖𝑖 = 1, 2, 3, 4 Then the probability of obtaining all 4 heads are Pr 𝐴𝐴 1 ∩ 𝐴𝐴 2 ∩ 𝐴𝐴 3 ∩ 𝐴𝐴 4 = Pr 𝐴𝐴 1 Pr 𝐴𝐴 2 Pr 𝐴𝐴 3 Pr 𝐴𝐴4 = 0.5 4 = 0.0625 Example: Roll a fair die 4 times. Let 𝐵𝐵 = 𝑠𝑠𝑜𝑜 𝑠𝑠𝑁𝑁𝑠𝑠𝑜𝑜𝑜𝑜 1 𝑜𝑜𝑖𝑖𝐴𝐴 𝑖𝑖𝑖𝑖 4 𝑁𝑁𝑜𝑜𝑠𝑠𝑠𝑠𝑜𝑜 Pr 𝐵𝐵 = 1 Pr 𝑖𝑖𝑜𝑜 𝑜𝑜𝑖𝑖𝐴𝐴 𝑖𝑖𝑖𝑖 4 𝑁𝑁𝑜𝑜𝑠𝑠𝑠𝑠𝑜𝑜 = 1 − � 𝐸𝐸=1 4 Pr 𝑖𝑖𝑜𝑜 𝑜𝑜𝑖𝑖𝐴𝐴 𝑜𝑜𝑖𝑖 𝑁𝑁𝑜𝑜𝑠𝑠𝑠𝑠 𝑖𝑖 = 1 5 6 4
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28 Section 2.3 Conditional Probability ( 條件概率 )
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