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# 1.1 - Solutions to Suggested Problems from Section 1.1...

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Solutions to Suggested Problems from Section 1.1 1.1.2 It’s easier to read the data off this table: word # letters # vowels suppose 7 3 a 1 1 word 4 1 is 2 1 picked 6 2 at 2 1 random 6 2 from 4 1 this 4 1 sentence 8 3 (a) 7 out of 10 words have at least 4 letters, 7 10 . (b) 4 out of 10 contain 2 or more vowels, 4 10 . (c) The same words counted in (b) all have 4 or more letters; 4 10 . 1.1.3 With replacement, the the number of pairs is n 2 . (a) The probabilitly of any one ordered pair is 1 /n 2 . (b) The number of ways this could happen is n - 1 (the first number could be any number 1 , . . . , n - 1, and the first number determines the second one), so the probability of this event is n - 1 n 2 (c) There are a couple of ways to see this one. The easiest is by symmetry: there are three things that can happen – either the first number is bigger than the second, the second is bigger than the first, or they are the same. The first two situations occur the same number of times (swapping the order of the numbers takes you from one situation to

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