Module 10 Written Assignment 1. 1 How many words can be created using all of the letters in the following state names? a. Florida b. Minnesota c. Wisconsin d. Tennessee a. b. c. d. 2. In how many ways can 24 cupcakes be divided among 11 birthday party guests, if every guest must receive at least one cupcake? 3. Use the Binomial Theorem to expand the following: 6 a. (a+ b) b. (x+ 1) 7 a. b. 4. Consider the following set a. b. c. d. {a , b , c , d } In how many different orders can this set be arranged? List all the possible arrangements of this set in dictionary order. What is the 15th permutation, that is, the 15th item on the list you created in step b? Write a Pseudocode algorithm for constructing any specified permutation of these four letters, without constructing or referring to the list. Verify that your algorithm correctly constructs the 15th permutation that you identified in part c. Hint: You may want to start with the example included at the end of this document, of a Pseudocode algorithm to identify the kth permutation of a set of 3 objects, and modify it to do 4 objects. Or you may do your own from scratch, if you prefer. This Module 10 Written Assignment 2 example is not the most efficient version, or the most general, but it does perform correctly in all cases. a. b. c. d. 5. A fair coin is tossed four times. a. List the entire sample space for this experiment. Use your sample space from part (a) to answer the following questions: b. What is the probability that it turns up heads every time? c. What is the probability of at least 2 heads? d. What is the probability of two tails in a row? a. b. c. d. 6. A single card is drawn from a standard 52-card deck. What is the probability of getting: a. A 10 or an Ace? b. An Ace or a diamond? EXAMPLE: Pseudocode Algorithm for Calculating kth Listed Permutation of 3 Elements Sample Calculations for k =5 Define List S S 1=a , S2=b , S 3=c S={a , b , c } Size = 3 (Starting value of Size is the number of elements in the original list.) * Calculate the total number of possible permutations of the elements of List S * Total = P * Input ordering number P3,3=6 k of desired permutation * Module 10 Written Assignment Perm = Input Value 3 k Perm = 5 Size Group = Total 6 3=2 Total = Group Count = 1 Do While Perm > Group (Perm = 5 > 2, so enter loop) Perm = Perm - Group Count = Count + 1 End Do FirstElt = Total = 2 1st execution 2nd execution Perm = 5 - 2 = 3 Perm = 3 - 2 = 1 Count = 1 + 1 = 2 Count = 2 + 1 = 3 (Perm = 1 not > 2, so exit loop) S Count FirstElt = S 3=c Define List S S 1=a , S2=b S={a , b } * Don't worry about how to redefine the list to leave out the selected element - it can be done. * Size = Size - 1 Size = 2 Size Group = Total Total = Group Count = 1 Do While Perm > Group Perm = Perm - Group Count = Count + 1 End Do Group = Total = 1 (Perm = 1 is not > 1, so do not enter loop) Since loop was not executed, Perm =1 and Count =1 S Count SecondElt = 2 2=1 SecondElt = S 1=a Define List S S 1=b S={b } Size = Size - 1 ThirdElt = S1 Size = 1 ThirdElt = S 1=b (the only element in a list of size 1) Define List L L1=FirstElt , L2=SecondElt , L3=ThirdElt Print L OUTPUT L= a. b. {c , a , b } Module 10 Written Assignment 4 Module 11 Written Assignment - Solutions 1 1. A fair die is rolled five times. a. What is the probability of getting five 6's in a row? b. What is the probability of getting all even numbers? 2. Network Reliability: In the network shown below, the links between computers can be either up or down. The reliability of each link is as shown in the diagram. The status of each link is independent of the status of the other links. What is the probability that there is a functioning set of links connecting computer B to computer C? A B Uptime = 85% Uptime = 93% Uptime = 96% C 3. Consider drawing one card from a standard 52-card deck. Define four events as follows: A={Hearts } , B={ cards } , C={ Aces } , and D={Queen of Hearts ,Queen of Spades } . Prove which pairs of these events are independent, and which are dependent. 4. An urn contains 7 red balls and 5 green balls. Two balls are drawn at random from the urn. What is the probability of getting two balls of the same color if: (a) the first ball is replaced before the second ball is drawn. (b) the first ball is not replaced before the second ball is drawn. Module 11 Written Assignment - Solutions 2 5. Communication Channel Reliability: Consider a communication channel over which a 0 or a 1 must be transmitted. Suppose that the probability that the bit to be sent is a 0 is 0.5, and the probability that a 1 is to be sent is also 0.5. Also suppose that due to noise the probability that a 0 is changed to a 1 during transmission is 0.06 and the probability that a 1 is changed to a 0 is 0.03. Suppose that a 0 is received. What is the probability that a 0 was sent