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Unformatted text preview: EL630: Homework 1
1. (Book; 21) Show that ( a ) A +B + A +B =A ; (b) ( A+B ) AB =AB +B A . (Hint: Use De Morgan’s Law A + B = AB, AB = A + B ) (Book: 22) If A = {2 ≤ x ≤ 5}and B = {3 ≤ x ≤ 6},find A + B, AB,and ( A + B ) AB . 2. 3. (Book: 23) Show that if AB = {∅}, then P ( A) ≤ P ( B ).
B A (Hint: In order to do this type of theoretical problems, you have to first understand the proofs of those four properties of probability in this lecture.)
4. (Book :2 − 4,) Show that (a ) if P ( A) = P ( B ) = P ( AB ), then P ( AB + B A) = 0; (b) if P ( A) = P ( B) = 1, then P ( AB ) = 1. 1 AB AB AB
(Hint: Try to first understand the proof of the last Theorem in Lecture 1, and notice that: A + B = AB + AB + AB, and AB, AB and AB are mutually exclusive.) 5. (Book :2 − 5) Prove thefollowing identity P ( A + B + C ) = P( A) + P ( B ) + P (C ) − P ( AB ) − P ( AC ) − P ( BC ) + P ( ABC )
A C B 6. A ⊂ B.Find P ( A + B ) and P ( AB )in terms of P ( A) and P ( B ).
B A 2 7. What is the probability that, in New York City, there are at least two people with the same number of hairs on the head? 8. Randomly cut a line segment into three pieces. What is the probability that these three pieces can form a triangle? 0 1 3 ...
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 Spring '10
 CHEN

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