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180ahw1 - ∪ B B ∩ C A ∩ B c and A ∩ B ∩ C c and...

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MATH 180A – Introduction to Probability Homework 1 (due 10/02/06) 1. A coin is tossed four times. Let A: exactly two heads B: heads and tails alternate C: first two tosses are heads (a) Which events, if any, are mutually exclusive? (b) Which events, if any, are subsets of other sets? 2. Compute P ( A c B c ) and P ( A c ( A B )) in terms of P ( A ), P ( B ) and P ( A B ). 3. Two dice are thrown. Let A: the sum is odd B: at least one die lands on an even number C: the sum is 7 Describe A B , A
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Unformatted text preview: ∪ B , B ∩ C , A ∩ B c and A ∩ B ∩ C c and compute the corresponding probabilities. 4. Derive P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C )-P ( AB )-P ( BC )-P ( AC ) + P ( ABC ) . *Bonus* Generalize to P ( ∪ n i =1 A i ) = X i P ( A i )-X i<j P ( A i A j ) + X i<j<k P ( A i A j A k )- ··· + (-1) n +1 P ( A 1 ··· A n ) . 5. Derive Boole’s inequality P ( ∪ i A i ) ≤ X i P ( A i ) . 1...
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