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20082ee131A_1_HW1SOL

20082ee131A_1_HW1SOL - 20 36 = 16 36 h Upon inspection in...

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EE 131A Homework #1 Solution Spring 2008 K. Yao 1. a. S = { S 2 , . . . , S 10 , SJ, SQ, SK, SA, H 2 , . . . , HA, D 2 , . . . , DA, C 2 , . . . CA } . b. P ( Ace of Spade ) = 1 / 52 . c. P ( Ace ) = 4 / 52 = 1 / 13 . d. P ( card 10) = 20 / 52 = 5 / 13 . e. For the “biased deck of cards,” we have: (b.) P ( Ace of Spade ) = 1 / 32 . (c.) P ( Ace ) = 4 / 32 = 1 / 8 . (d.) P ( card 10) = 4 / 72 + 16 / 32 = 1 / 18 + 1 / 2 = 10 / 18 = 5 / 9 . 2. a. There are 6 favorable outcomes, (3 , 4) , (4 , 3) , (6 , 1) , (1 , 6) , (5 , 2) , (2 , 5) , so prob is 6 / 36. b. There are 8 favorable outcomes, (6 , 1) , (5 , 2) , (4 , 3) , (1 , 6) , (3 , 4) , (2 , 5) , (6 , 5) , (5 , 6) so prob 8 / 36 . c. There are 6 outcomes where the two dice are equal. Of the remaining 30 outcomes, half have second > first. Answer is 15 / 36. d. Favorable outcomes are (6 , 1) , . . . , (6 , 6) , (1 , 6) , . . . , (6 , 6) , but don’t count (6 , 6) twice. An- swer is 11 / 36. e. Favorable outcomes are (5 , 5) , (5 , 6) , (6 , 5) , (6 , 6). Answer is 4 / 36. f. Upon inspection in the sample space, there are twenty such outcomes out of 36. Answer is 20 / 36 . g. Upon inspection in the sample space, there are sixteen such outcomes out of 36. P ( neither over 4) = 1 - P ( at least one
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Unformatted text preview: 20 / 36 = 16 / 36 . h. Upon inspection in the sample space, there are nine such outcomes out of 36. Answer is 9 / 36 . 3. The number of distinct ordered triplets with replacement = 7 × 2 × 52 = 728. 4. On the first day, there is a choice of 5 pairs, and subsequently, there is a choice of only 4 pairs. Total number is = 5 × 4 × 4 × 4 × 4 × 4 = 5 × 4 5 = 5120. 5. There are 4! = 24 ways in which the four volumes can be put back. Only one of these is correct. P (correct order) = 1 4! = 1 24 . 6. The total number of ways of arranging the word “cold” is 4! = 24. In considering the total number of ways of arranging the word “cool”, the letter “o” is repeated twice, thus 2! arrangements are the same. Thus, total number of ways of arranging the word “cool” is 4! / 2! = 24 / 2 = 12....
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