Practice Problems for Exam - First-Year Interest Group Seminar | N 1, Exams of Health sciences

Download Practice Problems for Exam - First-Year Interest Group Seminar | N 1 and more Exams Health sciences in PDF only on Docsity! 1. [5] Given sets A and B, each of cardinalityn ≥ 1 , how many functions map A in a one-to- one fashion onto B? 2. [5] a. Given the set of r symbols { , ,..., }a a ar1 2 , how many different strings of length n ≥ 1 exist? [5]b. Given the set of r symbols { , ,..., }a a ar1 2 , how many different strings of length n ≥ 2 exist that contain at least one a1 ? 3. [10] Present a combinatorial argument that for all positive integers x and y n k x y x yk n k k n n      = +− = ∑ 0 ( ) . (Hint: Consider sequences drawn from the union of distinct sets A and B of cardinalities x and y, respectively.) 4. [5] Given m a’s, n b’s, and p c’s, how many distinct sequences are there that employ each of the m n p+ + symbols exactly once? 5. [5] If four six-sided dice are thrown , how many different configurations (ignoring order) are possible? (Understand: the configuration {2, 2, 3, 4} is the same as the configuration {2, 3, 2, 4 } but different from {2, 3, 3, 4}) 6. [10] A pair of six-sided dice are thrown r times. (We ignore order in the pair but not in the sequence. Thus the sequence beginning (1,2), (2,6), (1,3),... is the same as the sequence (2,1), (2,6), (1,3),... but different from the sequence (2,6), (1,2), (1,3),...). How many such sequences have each of (1,1), (2,2), ..., and (6,6) appearing at least once within the sequence? (Hint: First figure how many configurations can be displayed in a single throw of the pair. Second, figure how many total sequences of length r there are, without considering the requirement for the 6 certain pairs. Lastly, let Ai = the set of such sequences avoiding (i,i) and apply the inclusion/exclusion principle.) 7. [5] a. Suppose k numbers are drawn from {1, 2, ..., n} allowing repetitions. Considering order to be relevant, how many such strings are there? [5] b. Now assume each of the different strings in part a. is equally likely. What is the probability that the maximum of the k numbers is greater than or equal to r, where 1≤ ≤r n ? 8. [5] Consider the following tables of probabilities for getting certain grades in a course and being in the freshman class. What is the probability of getting an A given that a student is a freshman? 9. [10] Suppose a message consists of 0’s and 1’s being transmitted with equal Freshman Non-Freshman A .3 .2 Less than A .4 .1 likelihood until five 1’s have been transmitted (i.e., the message terminates with the fifth 1). Give an expression for the expected value of the number of bits in the message? (Don’t waste time trying to simplify the expression.) 1. [5] Given set A of cardinality r ≥ 1 and set B, of cardinalityn ≥ 1 , how many functions map A in a one-to-one fashion into B? 2. [5] a. Given the set of r symbols { , ,..., }a a ar1 2 , how many different strings of length n ≥ 1 exist? [5]b. Given the set of r symbols { , ,..., }a a ar1 2 , how many different strings of length n ≥ 2 exist that contain at least two a1 ’s? 3. [10] Present a combinatorial argument that for all positive integers 1≤ ≤ ≤k m r : r m m k r k r k m k             =       − −       . 4. [5] Given m a’s, n b’s, p c’s and q d’s, how many distinct sequences are there that employ each of the m n p+ + + q symbols exactly once? 5. Consider sequences of the form < r r rn1 2, ,..., >, where the ri ≥ 0 . For fixed positive values of r and n, how many such sequences are there satisfying r r r rn1 2+ + + =... . 6. [5] a. Suppose k n≤ and k numbers are drawn from {1, 2, ..., n} without allowing repetitions. Considering order to be relevant, how many such strings are there? [5] b. Now assume each of the different strings in part a. is equally likely. What is the probability that the minimum of the k numbers is less than or equal to r, where 1≤ ≤r n ? 7. [5] Consider the following tables of probabilities for getting certain scores on an exam and being a CS major or not. Is the event of getting at lease a score of 80 independent of the event that the student is a CS major? CS Major non-CS Major 90 to 100 .08 .02 80 to 89 .17 .06 70 to 79 .28 .10 60 to 69 .14 .08 0 to 59 .08 .02 8. [10] Suppose a message consists of a total of n 0’s and 1’s being transmitted with equal likelihood. Give an expression for the expected value of the number of 1’s minus the number of 0’s in the message? (Don’t waste time trying to simplify the expression.) b. [10] Present a combinatorial argument that for all m and n satisfying 2 ≤ m , 2 ≤ n , and m n≤ +1: n m n m n m n m +      = +      + −       + −       2 1 1 2 (Hint: Consider A B c d= ∪ ∪{ } { } , where c d≠ , c B d B∉ ∉, , and #B n= .) 3. a. [10] For n ≥ 1 , how many decimal numbers between 1 and 10 1n − contain no 5's or 7's.? b. [10] For n ≥ 1 , how many decimal numbers between 1 and 10 1n − contain at most one 5 and one 7? 4. [10] For n ≥ 1 , how many ways can you find ordered triples ( , , )i j k so that i, j, and k are non-negative and their sum is n? (Hint: Consider balls and bins.) 5. [10] For n ≥ 1 , assume all strings of n digits from { , ,..., }0 1 9 are equally likely. What is the expected number of 9's in such a string? 6. [10] For n ≥ 3 , what is the probability that a string of length n of a's b's , c's, and d's has three or more a's (assuming all such strings are equally likely)? 7. [10] Consider two cards drawn from a 52 card deck and assume all such draws are equally likely. Is the event that a heart is drawn as the first card independent of the event that a spade is drawn of the second card? 1. [10] For ,4≥n consider a set A of cardinality n. How many subsets of A are of cardinality less than or equal to 3? 2. a. [10] Present a combinatorial argument that for all n and k satisfying n≤1 and nk ≤ : )!(!! knk k n n −⋅⋅      = b. [10] Present a combinatorial argument that for all positive values of n: ∑ − = += 1 0 212 n k kn (Hint: Consider Let k be the position of the first 1 in a bit string.) 3. a. [10] For n ≥ 1 , consider strings of length n from the set of characters {a, b, c, d, e, f}allowing repetition. How many such strings at most one a and one b? b. [10] For n ≥ 1 , how many decimal numbers between 1 and 10 1n − contain at most one 5 and one 7? 4. [10] For 1≥≥ nm , in how many ways can you place m balls into n boxes so that every box has at least one ball? 5. [10] For n ≥ 1 , assume all strings of n digits from { , ,..., }0 1 9 are equally likely. What is the expected number of 9's in such a string? 6. [10] For n ≥ 3 , what is the probability that a string of length n of a's b's , c's, and d's has three or more a's (assuming all such strings are equally likely)? 7. [10] Consider two cards drawn from a 52 card deck and assume all such draws are equally likely. Is the event that a heart is drawn as the first card independent of the event that a spade is drawn of the second card? 1. [5] Consider integers in the set {1, 2, 3, …, 1000}. How many are divisible by either 4 or 10? 2. a. [10] Present a combinatorial argument that for all 1≥n : nk n k k n 32 0 =     ∑ = b. [10] Present a combinatorial argument that for all nonegative integers p, s, and n satisfying nsp ≤+       +       + =      −       p sp sp n s pn p n (Hint: Consider choosing two subsets.) 3. [10] For n ≥ 1 , Let A = {1, 2, …, 2n}. How many subsets of A contain exactly 1k even numbers and 2k odd numbers? 4. [10] For 1≥n , how many ordered triples 1 2 3( , , )n n n of non-negative numbers satisfy nnnn =++ 321 ? (Hint: think about putting balls into bins.) 5. [10] For n ≥ 1 , assume all strings of n characters from { , , , }a b c d are equally likely. What is the expected number of a's in such a string? 6. [10] Given a finite event space E (in which all events are equally likely) and subsets A and B of E, show that 1)Pr()Pr()Pr( −+≥∪ BABA . 7. [10] Consider a 52 card deck of cards from which the ace of spades is removed resulting in a 51 card deck. Further, consider two distinct cards drawn from the 51 card deck and assume all such unordered draws are equally likely. Lastly consider the probability of the event that both cards are hearts. Is it more likely that both cards are hearts if it is given that both cards are face cards (i.e., Kings, Queens, or Jacks)? 8. [10] Let A be a set of cardinality p. Consider ordered strings of length m using the elements of A. How many such strings have the mth component a repetition of one of the preceding m-1? (Hint: Think about the complement and think about selecting the mth component first.) 1. [5] Suppose all rolls of a six-side die are equally likely. What is the probability the roll is a six given that it is not one? 2. a. [10] Present a combinatorial argument that for all 1≥n : 12 1 −=     ∑ = n n k k n (Note: The summation begins with 1=k .) b. [10] Present a combinatorial argument that for all integers k and n satisfying nk ≤≤3       − − +      − − +      − − +      − =      3 3 2 3 3 1 3 3 3 k n k n k n k n k n (Hint: Consider three special elements.) 3. [10] How many distinct permutations are there of the letters in “mississippi”? 4. [10] A bin has 100 blue balls, 100 red balls, and 100 green. How many different collections can I obtain using 100 of these balls? (Balls of the same color are indistinguishable from one another but are distinguishable from balls of another color. A “collection” has no order to it.) 5. [10] Assume all strings of length five using characters from { , , , }a b c d are equally likely. What is the probability that there is a substring abc in the string? 6. [10] Suppose a number k from {1, 2, …, 100} is to be drawn and that all numbers are equally likely. Let A be the event k is a power of two. Let B be the event k is an integer multiple of four. Prove either that the events A and B are statistically independent or that they are statistically dependent. 7. [10] For 1≥n , consider strings of length n containing 0’s and 1’s but ending in a 1. Assume all such strings are equally likely. What is the expected number of 1’s in such a string? 8. [10] For 1≥n , how many strings of length n employing the characters {a,b,c} have at least one a?