PS02_sol - University of Illinois Spring 2010 ECE 313...

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University of Illinois Spring 2010 ECE 313: Problem Set 2: Problems and Solutions Problem Set 2 Due: Wednesday February 3 at 4 p.m.. 1. [10 points] Each morning, before they go off to work in the mines, the seven dwarves line up and Snow White kisses each dwarf on the top of his head. In order to avoid any hint of favoritism, she kisses them in random order each morning. (a) What is the probability that the dwarf named Bashful gets kissed first on Monday? Solution: All dwarves have equal probability of getting kissed first on any day. Therefore, the probability that Bashful gets kissed first on Monday is 1 7 . (b) What is the probability that Bashful gets kissed first both Monday and Tuesday? Solution: The two events (Monday and Tuesday) are independent. Therefore, the probability that Bashful get kissed first both Monday and Tuesday is 1 7 · 1 7 = 1 49 . (c) What is the probability that Bashful does not get kissed first, either Monday or Tuesday? Solution: Bashful gets kissed first on either Monday or Tuesday. In other words, he can get kissed first on Monday and not on Tuesday or vice versa. Such probability is: 2 · 1 7 · (1 - 1 7 ) = 12 49 . (d) What is the probability that Bashful gets kissed first at least once during the week (Monday Friday)? Solution: The probability that Bashful does not get kissed first during the entire week is (1 - 1 7 ) 5 . Therefore, the probability that Bashful gets kissed first at least once during the week is 1 - (1 - 1 7 ) 5 . 2. [4 points] Alice and Bob roll a standard die obtaining a number at random from 1 to 6. What is the probability that Alices number is larger than Bobs number? What is the probability that Alice’s number is by one larger than Bob’s number? Solution: There are 6 · 6 = 36 possible number of events. The number of events where Alice’s number is larger than Bob’s number is 5 + 4 + 3 + 2 + 1 = 15 (each term corresponding to when Alice’s number is 6,5,4,3,2). Therefore, the probability that Alice’s number is larger than Bob’s number is 15 36 = 5 12 . On the other hand, the probability that Alice’s number is by one larger than Bob’s number is 5 36 = 5 36 .
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