# HEI - P N> 5 occurred what is the expected value of...

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University of Illinois Fall 2009 ECE 313: Hour Exam I Monday October 12, 2009 7:00 p.m. — 8:00 p.m. 100 Noyes Laboratory 1. [15 points] Let A , B , and C denote three events defined on a sample space Ω, and suppose that P ( A ) = 0 . 6 , P ( B ) = P ( C ) = 0 . 3, and P ( B c C ) = P ( A B c C c ) = 0 . 2. (a) [5 points] Find P ( B C ). (b) [5 points] Find P ( B C c ). (c) [5 points] Find P (( A B C ) c ). 2. [10 points] A and B are events defined on a sample space Ω. Assume P ( A ) , P ( B ) > 0. Mark each of the two statements below as TRUE or FALSE. No justification is needed. TRUE FALSE 2 2 If P ( A | B ) = P ( B | A ), then P ( A ) = P ( B ). 2 2 P ( A | B ) P ( B ) + P ( A c | B ) P ( B ) = P ( B ) . 3. [30 points] Especially in this problem, you must provide sufficient explanation to justify your numerical answers. A fair coin is tossed repeatedly until a Head occurs. N denotes the number of tosses. (a) [5 points] What is the expected value of N ? (b) [5 points] Find the numerical value of P { N > 5 } . (c) [10 points] Given that the event
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Unformatted text preview: P { N > 5 } occurred, what is the expected value of N ? (d) [10 points] Find the numerical value of E [cos( πN )]. 4. [20 points] A fair coin is tossed 10 times. Calculate the probability that the ﬁrst 5 tosses are all Tails given that a total of 8 Tails occurred on the 10 tosses. 5. [25 points] Dilbert has 3 coins in his pocket, 2 of which are fair coins while the third is a biased coin with P ( H ) = p 6 = 1 2 . The probability that a coin chosen at random from his pocket will land Tails is 7 12 . (a) [10 points] What is the value of p ? (b) [15 points] Dilbert picks two coins at random from his pocket, tosses each coin once, and observes a Head and a Tail. What is the conditional probability that both coins are fair?...
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