Probability and StatisticsUnit 3: Counting Principles

3.1 The Tree DiagramKA Video: “Count outcomes using a tree diagram”Notes from video:

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Practice:

1. At the after school Tiger Club meeting, there were four drinks you could choose from:  orange juice, Coke, Dr. Pepper, and water.  There were three snacks you could choose from:  peanuts, fruit, and cookies.  Each student may only have one drink and one snack.

A. Create a tree diagram showing all possible choices available.

B. Write the multiplication problem you could use to find the number of possible choices available.

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2. Tanya went shopping and bought the following items:  one red t-shirt, one blue blouse, one white t-shirt, one floral blouse, one pair of khaki capri pants, one pair of black pants, one pair of white capri pants, and one pair of denim shorts.

A. Make a tree diagram to show the choices she has to wear.

B. Write the multiplication problem you could use to find the number of possible choices available.

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3. Two dice are thrown together. Use a tree diagram to find the probability that one number is even and the other is

odd.

4. Two dice are thrown together. Use a tree diagram to find the probability that both numbers are less than five.

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5. Teddy has a two pairs of black shoes and three pairs of brown shoes. He also has three pairs of red socks, four pairs of brown socks and six pairs of black socks.

If Teddy chooses a pair of shoes at random and a pair of socks at random, what is the probability that he chooses shoes and socks of the same color?

(Hint: draw a tree diagram.)

6. Teddy has two pairs of black shoes and three pairs of brown shoes.

He also has three pairs of red socks, four pairs of brown socks and six pairs of black socks.

If Teddy chooses a pair of shoes at random and a pair of socks at random, what is the probability that the colors he chooses are black and brown?

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7. A bag contains 3 white balls, 4 green balls and 5 red balls. Three balls are drawn from the bag without replacement.

Use a tree diagram to find the probability that the balls are all of different colors.

8. A bag contains 3 white balls, 4 green balls and 5 red balls. Three balls are drawn from the bag without replacement.

Use a tree diagram to find the probability that the balls are all the same color.

Quiz grade: KA 3.1 (do at least 10)

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3.2 PermutationsKA Video A: “Factorial and counting seat arrangement”KA Video B: “Permutation formula”Notes from video(s):

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Practice:

Answer the following.  Show work.1.         A local restaurant offers a luncheon special consisting of a soup, a sandwich, and a beverage.  If a choice of three soups, two sandwiches, and four beverages is available, in how many ways can one select the luncheon special?

2.         How many different arrangements of the letters in the word math can be made?

3.         How many different ways can 6 students be seated in a row?

4.         How many 5-digit zip codes are possible if there cannot be any repeated digits?

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5.         The call letters of a radio station must have 4 letters.  The first letter must be a K or a W.  How many different station call letters can be made if repetitions are not allowed?  If repetitions are allowed?

6.         There are 10 true-false questions on a quiz.  How many different possible ways can someone answer all 10 questions?

7.         At a small high school, the principal wants to observe a teacher in each of the five hours of the school day.  If the school has 20 teachers, in how many ways can the principal select the five teacher observations?

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8.         How many different computer passwords are possible if each consists of 4 symbols; the first must be a letter and the other 3 must be digits?

9.         A Federal Express delivery route must include stops in four cities.  How many different routes are possible?

10.       Each social security number is a sequence of nine digits.  What is the probability of randomly generating nine digits and getting your own social security number?

11.       In a state, license plates consist of four letters and two digits.  How many different plates are possible?

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12.       From problem 11, how many different plates are possible if no repetitions are allowed?

13.       From problem 11, how many different plates are possible if only vowels and even numbers are to be used?

14.       A school’s Math Club has 15 participants.  In how many ways can they randomly select a President and Vice President?

15.       In a 10 person race, in how many ways can there be a first, second, and third place finisher?

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16.       A teacher notices after quizzes are turned in that three quizzes do not have a name written on them.  The teacher decides to randomly assign a quiz to each student.  What is the probability that the names assigned are the correct names for each student’s quiz?

17.       How many different arrangements of the letters in the word recess can be made?

18.       There are 6 contestants competing to be the fastest person to solve a Statistics problem.  In how many ways can there be a first, second and third place finishers.

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19.       Students at a local school are assigned a three letter pass code, where each letter, from A to Z is randomly assigned.  If each letter of the code must be different (you could NOT get a code “LOL”.), how many different pas codes are possible?

20.       In his refrigerator, Mr. Hemingway has 3 cans of Coke, 4 cans of Pepsi, and 2 cans of Mountain Dew.  He wishes to arrange them on one shelf.  In how many different ways can he arrange the 9 cans?  (Keep in mind, the 3 cans of Coke look identical; as do the 4 cans of Pepsi and the 2 cans of Mountain Dew.)

Quiz grade: KA 3.2 (do at least 10 problems)

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3.3 CombinationsKA Video A: “Introduction to Combinations”KA Video B: “Combination formula”Notes from video(s):

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Practice:

1. Katelyn asked the pet store owner for any four baby mice from a cage containing nine. How many possible combinations of mice could be picked?

2. There are 11 different marbles in a jar. How many different sets could you get by randomly picking 6 of them from the jar?

3. The volleyball team has nine players, but only six can be on the court at one time. How many different ways can the team fill the court?

4.  11 people were trying to be one of the first 5 callers to a radio station. How many different sets of people could have succeeded?

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5. A painter was carrying seven pails of different colored paint and dropped four of them, making a big mess. How many combinations of colors could he have spilled?

6. 5 names will be picked from a jar to be on a team. There are a total of 11 names in the jar. How many different combinations of names can be picked?

7. A number lock has 9 different digits. A combination of three digits can be set to open the lock. How many combinations are possible?

8. Bill has balls of 3 different sizes (small, medium and large) and in 5 different colors(white, green, red, yellow and blue). In how many possible combination can he choose two balls?

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9. A team of 3 students is to be formed out of 6 students. In how many ways can the teams be formed?

10. In a class, there are 15 students. Out of them, 9 are boys. A team is to be formed with 10 students of them 6 must be boys. In how many ways can the team be formed?

11. In how many ways can 3 pizza toppings be selected from a group of 12 toppings?

12. Find the number of permutations of the elements in the set {J, O, H, N}.

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13. Find the number of permutations of the elements in the set {S, O, R, R, Y}.

14. How many different 3-letter permutations can be formed using the letters in the word 'SCARED' exactly once?

15. How many permutations are there of the letters in the word CIRCUS?

16. How many 2-digit permutations can be formed using the 7 digits(0,1, 2, 3, 4, 5, 6)exactly once?

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17. How many outcomes are possible, when three coins are tossed?

18. What is the value of 3P6?

19.   In how many ways can you choose 2 even numbers and 3 odd numbers from the numbers 1, 2, 3, 4, 5, 6, 7?

20. What is the value of ?

Quiz grade: KA 3.3A, 3.3B

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3.4 Probability with Permutations and CombinationsKA Video A: “Probability through counting outcomes”KA Video B: “Probability using combinations”Notes from video(s):

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Practice:

Card Experiment – Probability1.   Find the probability of drawing a card that is red with an even numberfrom a full deck of cards.

2.   Find the probability of drawing a card that is black without a numberfrom a full deck of cards.

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Card Experiment – Permutations and Combinations

1.   List the permutations and combinations of selecting two cards from a set of five cards. Find the number of permutations and combinations.

Permutations List:

Number of permutations = __________

Combinations List:

Number of combinations = __________

2.   List the permutations and combinations of selecting three cards from a set of five cards. Find the number of permutations and combinations.

Permutations List:

Number of permutations = __________

Combinations List:

Number of combinations = __________

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Calculations with Formulas

3.   Determine the number of permutations and combinations for when we select 5 cards from a set of 20 cards. Show your work!

4.   Determine the number of permutations and combinations for when we select 5 cards from a set of 35 cards. Show your work!

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Probability Calculations

SPLIT THE DECK EVENLY AMONG EACH OF YOUR GROUP MEMBERS (each person should have a maximum of 13 cards) AND THEN ANSWER THE QUESTIONS BELOW.

7.   List the cards in your set of cards, such as 7 ♦ (diamonds), 2 ♣ (clubs) , J ♠

(spades), A♥ (hearts), etc.

8.   Find the probability of first picking a card with a club and then picking a card without a number on it, replacing the cards in the stack.

9.   Find the probability of first picking a card that is red suit and then picking a card that is red suit, without replacing the cards in the stack.

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10. Find the probability of picking a card with a number on it and picking a card without a number on it, without replacing the cards in the stack. (HINT: There are two calculations!)

11. Find the probability of first picking a card with an even number on it and then picking a card with an odd number on it, replacing the cards in the stack.

12. Find the probability of picking a card with a number less than 8 and picking a card with a number greater than 5, without replacing the cards in the stack. (HINT: There are two calculations!)

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13. Beth and Shayna each purchase one raffle ticket. If a total of eleven raffle tickets are sold and two winners will be selected, what is the probability that both Beth and Shayna win?

14. A meeting takes place between a diplomat and fourteen government officials. However, four of the officials are actually spies. If the diplomat gives secret information to ten of the attendees at random, what is the probability that no secret information was given to the spies?

15. A fair coin is flipped ten times. What is the probability of the coin landing heads up exactly six times?

16. A six-sided die is rolled six times. What is the probability that the die will show an even number exactly two times?

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17. A test consists of nine true/false questions. A student who forgot to study guesses randomly on every question. What is the probability that the student answers at least two questions correctly?

18. A basketball player has a 50% chance of making each free throw. What is the probability that the player makes at least eleven out of twelve free throws?

KA: 3.4A, 3.4B, 3.4C (10 in a row: CW grade), 3.4D (10 in a row: Quiz grade)

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