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Unformatted text preview: EE 351K Probability, Statistics, and Random Processes SPRING 2012 Instructor:S. Shakkottai firstname.lastname@example.org Homework 2 Solutions Problem 1 A hard disk storing information in binary form has been corrupted, so it can only be read with bit errors. Due to errors on the disk, a digit stored as 0 could be detected by you as 1 with probability 0.2; and a digit stored as 1 could be detected as 0 with probability 0.4. Each digit that is stored is a 1 or a 0 with equal probability. What is the probability that your detected value is the same as the stored value given that you detect a 0? Solution : Let A be the event that the stored value is 0, and let B be the event that the detected value is 0. The desired probability, P ( A | B ) , is found by Bayes rule: P ( A | B ) = P ( A ) P ( B | A ) P ( A ) P ( B | A ) + P ( A c ) P ( B | A c ) = . 5 . 8 . 5 . 8 + 0 . 5 . 4 . 67 . Problem 2 A parking lot consists of a single row containing n parking spaces ( n 2) . Alice arrives when all spaces are free. Bob is the next person to arrive. Each person makes an equally likely choice among all available spaces at the time of arrival. Describe the sample space. Obtain P ( A ) , the probability the parking spaces selected by Alice and Bob are at most 2 spaces apart, (i.e. there can be at most 1 spot vacant between them). Solution : For convenience, we will number each of the parking spaces. Alice can choose any of the n parking spaces. She has a probability of 1 /n of selecting any particular space. Bob can choose any of the remaining n- 1 spaces and has a probability of 1 / ( n- 1) of choosing any particular space (other than the one Alice chose)....
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