Chapter 4-4:
You want to obtain cash by using an ATM, but it's dark and you can't see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first. Complete parts 1 through 3. 1. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly? a) The probability is _____. b) This is a trick question. There is not enough information given to determine the answer. 2. What is the probability of randomly selecting the card's position and finding that it is incorrectly inserted on the first attempt, but is correctly inserted on the second attempt? (Assume that the same position used for the first attempt could also be used for the second attempt.) a) The probability is ______. b) This is a trick question. There is not enough information given to determine the answer. 3. How many random selections are required to be absolute sure that the card works because it is inserted correctly? a) The number of random selections required is _____. b) This is a trick question. There is no finite number of attempts, because it is possible to get the wrong positon every time.
1. a) The probability is 1/4. 2. a) The probability is 3/16. 3. b) This is a trick question. There is no finite number of attempts, because it is possible to get the wrong positon every time.
If the order of the items selected matters, then we have a __________.
If the order of the items selected matters, then we have a _permutation problem_.
A moving company has a truck filled for delieveries to ten different sites. If the order of the deliveries is randomly selected, what is the probability that it is the shortest route? P(randomly selecting the shortest route)=__________.
P(randomly selecting the shortest route)= 1/3628800 10!=3628800=1/3628800
When two basketball players are about to have a free-throw competition, they often draw names out of a hat to randomly select the order in which they shoot. What is the probability that they shoot free throws in alphabetical order? Assume each player has a different name. P(shoot free throws in alphabetical order)=_____.
P(shoot free throws in alphabetical order)= 1/2.
How many different ways can the letters of "grammar" can be arranged?
The number of different ways that the letters of "grammar" can be arranged is 630. 7!=5040 1!2!2!2!=8 5040/8=630
A classic counting problem is to determine the number of different ways that the letters of "occasionally" can be arranged. Find that number. The number of different ways that the letters of "occasionally" can be arranged is __________.
The number of different ways that the letters of "occasionally" can be arranged is 29937600. 12!=479001600 2!2!2!1!1!1!2!1!=16 479001600/16=29937600
A thief steals an ATM card and must randomly guess the correct four-digit pin code from a 5-key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first try? The number of possible codes is _____. The probability that the correct code is given on the first try is _____.
The number of possible codes is 625. 5^4=625 The probability that the correct code is given on the first try is 1/625
A Social Security number consists of nine digits in a particular order, and repetition of digits as allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt? The probability is _____.
The probability is 1/100000 10^4*10=100000
In a small private school, 4 students are randomly selected from 14 available students. What is the probability that they are the four youngest students? The probability is _____.
The probability is 1/1001 14C4=1001=1/1001
Winning the jackpot in a particular lottery requires that you select the correct four numbers between 1 and 58 and, in a separate drawing, you must also select the correct single number between 1 and 33. Find the probability of winning the jackpot. The probability of winning the jackpot is __________.
The probability of winning the jackpot is 1/14000910 58C4=424270 33C1=33 424270*33=14000910=1/14000910
With a short time remaining in the day, a delivery driver has time to make deliveries at 4 locations among the 9 locations remaining. How many different routes are possible? There are _____ possible different routes.
There are 3024 possible different routes. 9P4=3024
If the radio station call letters must begin with either K or W and must include either two or three additional letters, how many different possibilities are there? There are _____ different possibilities.
There are 36504 different possibilities. 2(26^2+26^3)=36504
In a state pick 4 lottery game, a bettor selects four numbers between 0 and 9 and any selected number can be used more than once. Winning the top prize requires that the selected numbers match those and are drawn in the same order. Do the calculations for this lottery involve the combinations rule or either of the two permutations rules? Why or why not? If not, what rule does apply? a) The combination rule applies to this problem because the numbers are selected with replacement. Neither of the permutations rules allows replacement. b) The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement. The multiplication counting rule applies to this problem. c) The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement. The factorial rule applies to this problem. d) The permutation rule (with different items) applies to this problem because repetition is allowed. The permutation rule (with some identical items) and the combination rule cannot be used iwth repetition. e) The permutation rule (with some identical items) applies to this problem because repetition is allowed. The permutation rule (with different items) and the combination rule cannot be used with repetition.
b) The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement. The multiplication counting rule applies to this problem.
In horse racing, a trifecta is a bet that the first three finishers in a race are selected, and they are selected in the correct order. Does a trifecta involve combinations or permutations? Explain. a) Because the order of the first three finishers does not make a difference, the trifecta involves permutations. b) Because the order of the first three finishers does make a difference, the trifecta involves combinations. c) Because the order of the first three finishers does make a difference, the trifecta involves permutations. d) Because the order of the first three finishers does not make a difference, the trifecta involves combinations.
c) Because the order of the first three finishers does make a difference, the trifecta involves permutations. Permutations: an arrangement of all or part of a set of objects, with regard to the order of the arrangement.
Which of the following is NOT a requirement of the Combinations Rule, nCr=n!/r!(n-r)!, for items that are all different? a) That r of the n items are selected (without replacement). b) That order is not taken into account (consider rearrangements of the same items to be the same). c) That order is taken into account (consider rearrangements of the same items to be different sequences.) d) That there be n different items available.
c) That order is taken into account (consider rearrangements of the same items to be different sequences.)
A combination lock uses three numbers between 1 and 65 with repetition, and they must be selected in the correct sequence. Is the name of "combination lock" appropriate? Why or why not? a) No, because the permutations rule would be used to determine the total number of combinations. b) Yes, because the combinations rule would be used to determine the total number of combinations. c) No, because factorials would be used to determine the total number of combinations. d) No, because the fundamental counting rule would be used to determine the totalnumber of combinations.
d) No, because the fundamental counting rule would be used to determine the totalnumber of combinations.
Which of the following is NOT a requirement of the Permutations Rule, nPr=n!/(n-r)!, for items that are all different? a) Order is taken into account (rearrangements of the same items are considered to be different). b) Exactly r of the n items are selected (without replacement). c) There are n different items available. d) Order is not taken into account (rearrangements of the same items are considered to be the same).
d) Order is not taken into account (rearrangements of the same items are considered to be the same).