s1 - Math 4653: Elementary Probability: Spring 2007...

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Math 4653: Elementary Probability: Spring 2007 Homework #1. Problems and Solutions 1. Appendix 1 (vi): Prove that ± 2 n n ² = n X k =0 ± n k ²± n n - k ² = n X k =0 ± n k ² 2 . Solution. The left side is the number of all subsets of the set { 1 , 2 , . . . , n - 1 , n, n +1 , . . . , 2 n } , which consist of n elements. We can count this number as follows. For each k = 0 , 1 , . . . , n , there are ( n k ) possible ways to choose k elements from { 1 , 2 , . . . , n - 1 , n } , and independently, there are ( n n - k ) = ( n k ) ways to choose the remaining n - k elements from { n + 1 , . . . , 2 n } . By the multiplication rule, there are ( n k )( n n - k ) ways to choose n elements from { 1 , 2 , . . . , n, n +1 , . . . , 2 n } , such that exactly k are taken from { 1 , 2 , . . . , n - 1 , n } . Finally, in order to find the total number with arbitrary k , we need to do summation with respect to k . 2. Appendix 1 (x): How many different eleven-letter words (not necessarily pronounceable or meaningful!) can be made from the letters in the word MISSISSIPPI? Solution. If all eleven letters were distinct, then there would be 11! ways to arrange the letters. However, there are 4! ways to rearrange the four identical ”i”’s, another 4! ways to rearrange the four identical ”s”’s, and 2! ways to rearrange two identical ”p”’s, which yields the answer 11! 4! · 4! · 2! = 34650 . 3. Find the number of nonnegative integer solutions of the inequality x 1 + x 2 + x 3 + x 4 4. Hint: Use the analogy with the number of ways to place n identical balls into k different boxes, which is ( n + k - 1 n ) . Solution. The given inequality is equivalent to the equality x 1 + x 2 + x 3 + x 4 + x 5 = 4 with x 5 = 4 - ( x 1 + x 2 + x 3 + x 4 ). The number of solutions is the same as the number of ways to place n = 4 identical balls into k = 5 different boxes, which is ± 8 4 ² = 70 . 4. Sec. 1.1, #2: Suppose a word is picked at random from this sentence. Find: a) the chance that the word has at least 4 letters; b) the chance that the word contains at least 2 vowels (a, e, i, o, u); c) the chance that the word contains at least 4 letters and at least 2 vowels. Solution.
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s1 - Math 4653: Elementary Probability: Spring 2007...

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