Name: _____________________________________________________________________________ North Suburban Mathematics League—Counting Basics & Simple Probability— With Calculator! ***************************************** Frosh 78-79 ************************************* 1) The following table represents a survey of men and women of two age categories to see if they are Democrats, Republicans, or Independents.

Democrat Republican Independent

Men over 25 30 25 15

Men not over 25 35 5 10

Women over 25 20 30 5

Women not over 25 15 15 5

What is the probability that a person selected at random will be a Republican?

2) Use the table from problem #1. What is the probability that a person over 25 is a Republican?

3) Use the table from problem #1. If the probability of a republican drinking tea is assumed to be 14 probability of a Democrat drinking tea is 52 , and the probability of an Independent drinking tea is 12 , then what is the probability that a person selected at random will be a tea drinking Republican?

4) Use the table from problem #1 and the information from problem #3. What is the probability that a person selected at random will be a non-tea drinker?

5) Al and Bill are married to Clara and Doris, but not necessarily respectively. They have children Ed, Fred, and George. One family has 2 children and the other only one. If a father, a mother, and two children are selected at random, what is the probability that they are all in the same family?

********************************** Frosh 88-89 ******************************************** 6) Tom is tossing a fair coin. He tosses it up once and it lands “heads.” What is the probability it will land “heads” on the next toss?

7) If Tom were to toss the coin three consecutive times, what is the probability that it would land with the heads side up all three times?

8) Two standard dice are rolled. What is the probability that the sum of the numbers rolled is even?

9) A mathematics team has a total of six students, including two “experts,” who can participate in the Probability competition. The coach is late for the meet and three students are selected at random to be the ones whose scores will count for the team total. What is the probability that exactly one of the “experts” will have his score included in the team total? Report answer as a reduced fraction.

10) Assume the probability of the Bears winning against the Vikings is 0.6. If the two teams play 4 times over the season, what is the probability that the Bears will win at least 3 of the games? Report answer as a decimal rounded to four places.

********************************* Frosh 89-90 ****************************************** 11) Suppose there are three cards, one with red dots on the front and back, one with blue dots on the front and back, and one with a red dot on one side and a blue dot on the other side. A card is selected at random, and a red dot is observed on the face. What is the probability that there’s a red dot on the back?

12) In a standard dice game, using two dice, a player wins on the first roll if he rolls a sum of “7” or “11.” What is the probability of winning on the first roll?

13) Ray Ovack has an old flashlight with a good switch. However, the flashlight only works if the bulb is “good” and the batteries are “good.” The probability that the bulb is “not good” is 0.089. The probability that the batteries are “not good” is 0.123. If Ray turns the switch on, find the probability that the flashlight will NOT work. Round answer to nearest thousandth.

14) In a LOTTO game, three “digits,” 0 through 9, are selected at random. What is the probability that the three digit number is even or divisible by three? (Note: digits can repeat and numbers such as “007” are possible)

15) Assume that a basketball team consists of two tall centers and eight other players, for a total of ten. To be fair, the coach says that he will select the starting five players at random. What is the probability that the starting team will have at least one of the two tall centers?

******************************* Frosh 91-92 ******************************************** 16) With one octave, “do-re-mi-fa-sol-la-ti-do,” how many 3 note chords can be formed? To play a chord, one plays the three notes simultaneously. (Of course some chords would sound awful!)

17) How many ways can the word TRIANGLE be misspelled, if all of the letters are used, and the only thing the speller knows is that it begins with a “T” ?

18) How many different positive integers, with no repetition of digits and no initial digit of “0,” can be formed with the digits from the set 0,1, 2,3, 4 ?

19) In how many ways can 6 different candy bars be distributed to 3 kids, if each one is to receive two candy bars?

20) An executive board has eight members. From these eight, three officers, a president, a secretary, and a treasurer must be chosen. Two of the members, Abby and Beau, will not serve together as officers. If this last condition is agreed to, how many ways can three officers be chosen from the eight members?

********************************** Frosh 95-96 ***************************************** 21) Two different numbers x and y are randomly selected from the set 1, 2,3, 4,5 . What is the probability that the sum x  y is odd?

22) You have a standard die and you are playing against an opponent who also has a standard die. You each roll your die once. You win if your die shows a higher number than your opponent. What is the probability that you win the game?

23) When an arrow randomly hits a circular target, the probability of hitting the bull’s-eye is 0.5. If the radius of the bull’s-eye is 6 inches, what is the circumference of the target? Report answer to nearest hundredth of an inch.

24) In poker, a straight is a group of cards in consecutive rankings. The rankings are A-K-Q-J-10-9-8-7-6-54-3-2-A. An ace can only be at the end of a sequence (either high or low). The suits of the cards do not matter. For instance, a 3-card straight could be a 2 of clubs, 3 of hearts, and ace of spades. There are 270,725 ways to be dealt a 4-card hand from a deck of playing cards. If you are randomly dealt a 4-card hand, what is the probability, to the nearest millionth, that those cards form a 4-card straight?

25) Consider the lattice points on a standard coordinate plane. A lattice point is an intersection point on the grid. A “move” is going exactly one unit north, south, east, or west. Suppose you start at the origin. If you do 4 random moves, what is the probability that you end up at 1,1 ?

***********************************Frosh 97-98 ****************************************

26)

If you had 10 lines, how many regions would you have?

27) Pascal’s Triangle is shown below on the left. Each outer term is a “1” and each interior term is found by adding the two terms above. The triangle to the right is formed by the same procedure, except the end terms are 1’s and 2’s. What is the sum of the terms in the 12th row from the top in the triangle on the right?

28) In the following sequence, the first three terms are the first three positive odd integers, the next three terms are the first three positive even integers, the next three terms are the second through fourth positive odd integers, and the next three are the second through fourth positive even integers. This pattern continues. What is the sum of the first 99 terms of the sequence? 1, 3, 5, 2, 4, 6, 3, 5, 7, 4, 6, 8, 5, 7, 9, …

29) Three red and nine white chips are placed in a circle. In how many ways can they be placed so that two red chips are never next to each other?

30) Notice that 9  9  18 and 9  9  81. Notice that 3  24  27 and 3  24  72 Find another pair of numbers that fit this pattern.

********************************* Frosh 99-00 ****************************************** 31) A positive integer between 1 and 100 inclusive is selected at random. What is the probability that the integer selected and 12 have no common factors other than 1?

32) The newspaper says there is a 1 in 3 chance of rain for the Saturday football game. Mark wants to go to the game, so he says that since it is a 1 in 3 chance of rain, he will roll a die. A 1 or 2 means he thinks it will rain, so he will stay home. A 3, 4, 5, or 6 means he thinks it will not rain so he will go to the game. If we assume the newspaper’s probability of rain on Saturday is correct, what is the probability that mark will go to the football game and get rained on?

33) Each school can enter six frosh in a contest, and the top four scores count for the schools total. At Sowdim High School, the six contestants are completely unprepared so they will simply guess at answers, and the order of the six scores is completely random, and no ties will occur. To impress his mother, Arthur says that it is likely that he will be one of the four scores counted for Sowdim. What is the probability that his score will count?

34) Using a 3 by 3 grid as shown, three X’s are placed randomly in three of the squares. What is the probability that these three X’s are in a row (horizontally, vertically, or diagonally)?

35) In how many ways can four different gifts be given to three different people? For example, all four gifts can be given to one person.

************************************* Frosh 01-02 **************************************

36) At the opening of the U.S. Supreme Court session, the first judge will enter, then the second judge will enter and shake his hand. The third judge will then enter and shake the hands of these two, and the process will continue until the ninth judge, the Chief Justice, will enter and shake the hand of all the judges. If each handshake takes 10 seconds, how long will the ceremony take? Report answer in minutes.

37) Each of the seven dwarfs wants to kiss Snow White before they go off to work. It is decided that Doc, will always give the first kiss and Grumpy will never give the final kiss. Given those two conditions, in how many ways can the seven dwarves line up in a row to kiss Snow White good-bye?

38) A coin is unbalanced so that 60% of the time it will land heads. That coin, and two other normal coins are tossed. What is the probability that two of the coins will be heads and the other tails?

39) In the coordinate plane, a lattice point is an ordered pair in which the x and y coordinates are both integers. Consider the array of lattice points in which the coordinates are such that 0  x  3 and 0  y  3 . How many squares can be formed for which the vertices are lattice points in that array?

40) In the figure, there are no missing cubes that are not seen. That is, each stack of cubes reaches the base. The top level of this stack has one cube, the next level downward has three cubes, the next has six cubes, and so on. If this process were continued until there are 11 levels, how many cubes in all would be needed to build the structure?

*********************************** Frosh 02-03 ******************************************* 41) A class has 10 males and 14 females. Two students are selected at random. What is the probability that both are males?

42) In a standard deck of 52 cards, there are four suits (clubs, diamonds, hearts, spades) and 13 cards in each suit (2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A). If a card is chosen at random, what is the probability that it is an ace or a spade?

43) 1000 people who saw both Harry Potter and The Lord of the Rings were asked to rate the movie. 700 said they liked Harry Potter, 600 said they liked The Lord of the Rings, and 400 said they liked both. A person from the reviewing group is selected at random. What is the probability that this person did not like either movie?

44) Assume that next year the Cubs and the Sox play in the World Series, and that the likelihood of either team winning a single game is 12 . In the World Series, the first team to win four games out of a possible seven is the world champion. Suppose that in the first three games, the Cubs win two and the Sox win one. What is the probability that the Sox will still win the World Series?

45) Assume there are 15 cards, one each with each whole number from 1 to 15, and three are drawn at random, without replacement. What is the probability that the sum of the three numbers is equal to 20?

************************************ Frosh 04-05 ***************************************

46) In how many ways can the five letters of the word LATIN be arranged in a row?

47) Ten balls, identical in size, are placed in a bag. Six are red and four are blue. Two balls are withdrawn, one after the other, without replacement. What is the probability that they are both of the same color?

48) A nickel is tossed in the air three times. It lands on the floor. What is the probability that all three outcomes are the same (that is three heads or three tails)?

49) GOCHCIA is one anagram of the word CHICAGO. How many distinguishable anagrams of CHICAGO, including CHICAGO itself, are there?

50) Draw one card from a standard 52-card deck. If it is a heart, stop. If not, put the card back in the deck, shuffle, and draw again, replacing and shuffling if you fail to get a heart. Keep drawing until you get a heart. What is the probability that it will take you at least four draws to get the first heart?

************************************ Frosh 05-06 ***************************************

51) Ms. Germaine has five students: Aleksandr, Ceasar, Mili, Modest, and Nikolai. If she promises Mili that he is first in line and Modest that he is last in line, how many ways can she line up her students?

52) Jack and Jill want to find the probability of getting a head and two tails when a coin is flipped three times. Jack finds the probability of getting a head and two tails in any order, while Jill finds the probability of getting a head first and then two tails after that. What is the difference when you subtract their answers (Jack’s answer minus Jill’s answer)?

53) How many ways are there to divide six people into two teams of three people each? It does not matter which team is which.

54) A baseball team has 14 hitters and 11 pitchers. If two players are chosen at random to form a pair, what is the probability of choosing one player of each type? Express your answer as a simplified fraction.

55) Two spinners each have the whole numbers 1 through 100 inclusive, so that each number has a 0.01 probability of occurring on a single spin of either spinner. If both spinners are spun one after the other, what is the probability that the second spinner will point to a greater number than the first spinner? Express your answer as a simplified fraction.

********************************** Frosh 07-08 ***************************************** 56) There are four girls with whom Mark can go to the pep rally, but not to the dance, and there are three girls with whom he can go to the dance, but cannot take to the pep rally. There are also two other girls who are always jealous, and if he goes to the pep rally with one of them, he must also go with her to the dance. How many ways can he have dates for both the pep rally and the dance?

57) A town has all streets in a rectangular grid, running either north and south, or east and west, and all parallel streets are exactly one block apart. If Mark lives three blocks south and four blocks west of Marian, how many paths can he take from his house to hers, if he only travels east or north?

58) A club has 10 members, five male and five female. In how many ways can an executive committee of four members be chosen if at least one has to be male?

59) Alfred randomly selects a digit from 1 through 9, inclusive, then Betty randomly selects a digit from 1 through 9, inclusive. A fraction is then created with the digit Alfred selected used for the numerator, and the digit Betty selected for the denominator. What is the probability that the fraction is greater than 1?

60) Urn I has 3 red balls and 9 white balls, urn II has 5 red balls and 5 white balls, and urn III has 7 red balls and 5 white balls. One of the three urns is selected randomly and a ball is drawn. What is the probability that the chosen ball is red?

**********************************Frosh 09-10 ******************************************

61) A series of squares are arranged in columns as shown. If the last column of squares is 1000 squares in height, how many squares are there in total?”

62) How many different positive 8-digit numbers can be formed using each of the digits from 1,2,3,4,5,5,6,6

63) How many positive integers less than or equal to 1000 are multiples of 2, 3, or 5?

64) Given a 15-gon (a polygon with 15 sides), how many triangles can be formed using the diagonals and at least one side of the polygon?

65) You are given 5 different consonants and 3 different vowels. From all possible six-letter words using a letter from the list at most once, what is the probability that a word chosen at random will have a consonant as the first letter, and will contain at least two vowels?

*********************************** Frosh 10-11 **************************************** 66) A fair coin is tossed and it lands “heads” three consecutive times. If the coin is tossed again, what is the probability it will land as “heads?”

67) There is a dice game in which five dice are rolled. Each die is a 6-sided cube. If all five dice have the same number showing, this is called a “Yahtzee.” What is the probability that you can roll a “Yahtzee” with a single roll of the five dice? Report answer as a reduced rational fraction.

68) The diagram shows the home of Argyle and his girlfriend Beatrice. If Argyle wants to visit Beatrice, he can walk north (up) or east (right). How many different routes can Argyle take from his home to Beatrice’s, only walking north and east?

69) A bag contains 4 red balls, 3 blue balls, and 3 green balls. Three balls are simultaneously drawn at random. What is the probability that they are all the same color?

70) Refer to problem #68 above. Assume Argyle is not very bright and whenever he gets to a corner, he randomly turns north or east. What is the probability he will NOT get to Beatrice’s house? (in other words, what is the probability that he will walk off the grid?)

************************************ Frosh 11-12 ****************************************** 71) How many different 4-letter “words,” some not pronounceable, can be made from the letters A, F, M, R, using each letter once?

72) How many different 4-letter “words,” some not pronounceable, can be made from the letters A, C, O, S, T, if letters may be used more than once?

73) Two cards are drawn, without replacement, from a standard deck of cards. What is the probability that both are spades?

74) Two dice are rolled and the sum is recorded. What is the probability that the sum will be at least 9?

75) Ann, Ben, Carl, David, Emily, and Franklin are to pose in a row for a photograph. If Ben and Franklin refuse to stand next to each other, how many arrangements are possible?

************************************ Frosh 12-13 ****************************************** 76) In a list of all positive integers from 1 through 1000, how many times does the digit “0” occur?

77) Tom has 5 mathematics books, 4 physics books, and 3 history books. In how many ways can he arrange the books on a shelf if the books for each subject must be together?

78) In a state, license plates have 1, 2, or 3 letters followed by 4 digits. No letter may be repeated, but digits may be repeated. How many different license plates are possible?

79) Consider a cube. If each of the eight corners are cut, it will produce a triangular face, and what remains are six octagons where there were once squares. This is how a “truncated” cube is formed. Note that at each vertex, two octagons and a triangle meet. In all, how many vertices and edges does a truncated cube have? Give your answer as the sum of the vertices plus the edges.

80) Using only pennies, nickels, dimes, and/or quarters, how many ways can there be exactly $0.50?

************************************ Frosh 13-14 ****************************************** 81) There are 8 appetizers, 22 main courses, and 6 desserts on a menu. How many ways can a customer select a meal consisting of one of each of the above?

82) There is a group of four girls and four boys. If order does not matter, how many ways can two people be chosen who are of the same gender?

83) You want to form numbers that are ten digits long. How many of these contain eight “5’s” and one “6” and one “7?”

84) Students A, B, C, D, E, and F need to line up from left to right. Students D and F must be immediately next to each other. How many ways can the students arrange themselves?

85) Three standard six-sided dice are tossed. What is the probability that the numbers shown sum up to 6?