20065ee131A_1_EE131AF06HW1 - events i.e P a P b and P c...

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Assignment 1, EE 131A Due: Wednesday October 11, 2006 Problem 1. Two components in a system, C 1 and C 2 are tested and declared to be in one of the three possible states: F , functioning; R , not functioning but repairable; and K , kaput. (a) What is the sample space in this experiment? (b) What is the set corresponding to the event “none of the components is kaput”? Problem 2. Let A , B , and C be events in a sample space S . Use the set theoretic operations (i.e., union, intersection, and complement) to express the following events: (a) Exactly one of the three events occurs. (b) Exactly two of the events occur. (c) One or more of the events occur. (d) Two or more of the events occur. (e) None of the events occur. Problem 3. A random experiment has a sample space S = { a,b,c } . Suppose that P [ { a,c } ] = 5 8 and P [ { b,c } ] = 7 8 . Use the axioms of the probability to find the probabilities of the elementary
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Unformatted text preview: events; i.e., P [ a ] , P [ b ] , and P [ c ] . Problem 4. Show that (a) P [ A ∪ B ∪ C ] ≤ P [ A ] + P [ B ] + P [ C ] . (b) If A ∩ B = ∅ then P [ A ] ≤ P [ B c ] . Problem 5. How many seven-digit telephone numbers are possible if the first number is not allowed to be 0 or 1? Problem 6. Ordering a “deluxe” pizza means you have four choices from 15 available toppings. How many combinations possible if toppings can be repeated? If they cannot be repeated? Problem 7. A deck of cards contains 10 red cards numbered 1 to 10 and 10 black cards numbered 1 to 10. How many ways are there of arranging the 20 cards in a row? Suppose we draw the cards at random and lay them in a row. What is the probability that red and black cards alternate?...
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