A02ans - EXERCISES IN STATISTICS Series A, No. 2 1. Let A1...

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EXERCISES IN STATISTICS Series A, No. 2 1. Let A 1 , A 2 be subsets of a sample space S . Show that P ( A 1 A 2 ) P ( A 1 ) P ( A 1 A 2 ) P ( A 1 )+ P ( A 2 ) . Answer: (i) To prove that P ( A 1 A 2 ) P ( A 1 ) we take P ( A 1 A 2 )= P ( A 2 | A 1 ) P ( A 1 ) Dividing throughout by P ( A 2 | A 1 ) gives P ( A 1 P ( A 1 A 2 ) P ( A 2 | A 1 ) P ( A 1 A 2 ) since P ( A 2 | A 1 ) 1 . (ii) To prove that P ( A 1 ) P ( A 1 A 2 ), we take P ( A 1 A 2 P ( A 1 { P ( A 2 ) P ( A 1 A 2 ) } . But P ( A 2 ) P ( A 1 A 2 ) 0so P ( A 1 ) P ( A 1 A 2 ) . (iii) To prove that P ( A 1 A 2 ) P ( A 1 P ( A 2 ) we take P ( A 1 A 2 P ( A 1 P ( A 2 ) P ( A 1 A 2 ) and we simply note that P ( A 1 A 2 ) 0 . 2. Find the probabilities P ( A ), P ( B ) when A , B are statistically independent events such that P ( B )=2 P ( A ) and P ( A B )=5 / 8. Answer: The assumption of independence indicates that P ( A B ) = P ( A ) P ( B ). Using this, and then the fact that P ( B P ( A ), we ±nd that P ( A B P ( A P ( B ) P ( A B ) = P ( A P ( B ) P ( A ) P ( B ) = P ( A )+2 P ( A ) 2 { P ( A ) } 2 = 5 8 .
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A02ans - EXERCISES IN STATISTICS Series A, No. 2 1. Let A1...

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