hw1sol

# hw1sol - UCSD ECE153 Handout#8 Prof Young-Han Kim Thursday...

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Unformatted text preview: UCSD ECE153 Handout #8 Prof. Young-Han Kim Thursday, April 7, 2011 Solutions to Homework Set #1 (Prepared by TA Yu Xiang) 1. World Series. The World Series is a seven-game series that terminates as soon as either team wins four games. Suppose San Diego Padres (denoted as A) and New York Yankees (denoted as B) match up in the series. Possible outcomes include AAAA, ABABABA, and BBBAAAA. Assume that each game is independent and both teams are equally strong. (a) Describe the sample space of all possible outcomes. (b) What is the probability that Padres will win the series? (c) What is the probability that all seven games will be played? (d) Suppose Padres lost the first three games. What is the (conditional) probability that they will still win the series? Solution: (a) The sample space consists of two situations: Padres win or Yankees win. First we consider the outcomes that Padres win the game. There are five possible cases. i. Padres don’t lose at all. AAAA There is only one outcome for this case. P { SD 4 - NY 0 } = 1 2 4 = 1 16 . ii. Padres lose one game. BAAAA, ABAAA, AABAA, AAABA In this case, there are five games and Padres must win the last game (why), Among first 4 games we can have arbitrary combination of 3A1B, so there are ( 4 1 ) = 4 outcomes. P { SD 4 - NY 1 } = 4 2 5 = 4 32 = 1 8 . iii. Padres lose two games. BBAAAA, ABBAAA, AABBAA, AAABBA, BABAAA ABABAA, AABABA, BAABAA, ABAABA, BAAABA There are ( 5 2 ) = 10 outcomes for this case. P { SD 4 - NY 2 } = 10 2 6 = 10 64 = 5 32 . 1 iv. Padres lose three games. BBBAAAA, ABBBAAA, AABBBAA, AAABBBA, BBABAAA ABBABAA, AABBABA, BABBAAA, ABABBAA, AABABBA BBAABAA, ABBAABA, BBAAABA, BAABBAA, ABAABBA BAAABBA, ABABABA, BABABAA, BAABABA, BABAABA There are ( 6 3 ) = 20 outcomes for this case. P { SD 4 - NY 3 } = 20 2 7 = 20 128 = 5 32 . Vice versa, if Yankees win, the cases would be the same. So we can write our sample space as: Ω = AAAA BAAAA,ABAAA,AABAA,AAABA BBAAAA,ABBAAA,AABBAA,AAABBA,BABAAA ABABAA,AABABA,BAABAA,ABAABA,BAAABA BBBAAAA,ABBBAAA,AABBBAA,AAABBBA,BBABAAA ABBABAA,AABBABA,BABBAAA,ABABBAA,AABABBA BBAABAA,ABBAABA,BBAAABA,BAABBAA,ABAABBA BAAABBA,ABABABA,BABABAA,BAABABA,BABAABA BBBB ABBBB,BABBB,BBABB,BBBAB AABBBB,BAABBB,BBAABB,BBBAAB,ABABBB BABABB,BBABAB,ABBABB,BABBAB,ABBBAB AAABBBB,BAAABBB,BBAAABB,BBBAAAB,AABABBB BAABABB,BBAABAB,ABAABBB,BABAABB,BBABAAB AABBABB,BAABBAB,AABBBAB,ABBAABB,BABBAAB ABBBAAB,BABABAB,ABABABB,ABBABAB,ABABBAB ....
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hw1sol - UCSD ECE153 Handout#8 Prof Young-Han Kim Thursday...

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