Worksheet C: Pascal’s Triangle



Geometry & Finite X

Unit 15 – Probability & Combinatorics

| | | |

|DAY |TOPIC(S) |ASSIGNMENT |

| |13.1 Sample Spaces and The Fundamental Counting |Page 902 - 904 {3, 9, 15 – 18, 20, 23} |

|1 |Principle | |

| |13.2 Permutations and Factorials |Worksheet A |

|2 | | |

| |13.2 Special Permutations |Worksheet B |

|3 | | |

| |13.2 Combinations |Worksheet C |

|4 | | |

| |Review Counting Principles |Worksheet D |

|5 |(Mixed Problems) | |

| |Quiz 13.1 and 13.2 |Worksheet E: Exam Review |

|6 | | |

| |13.4 Intro to Probability and Simulations |Worksheet F |

|7 | | |

| |13.6 Probability of Mutually Exclusive Events and |Worksheet G |

|8 |the Addition Rule | |

| |13.5 Probability of Independent Events and the |Worksheet H |

|9 |Multiplication Rule | |

| |Probability Practice: Independent or |Worksheet I |

|10 |Mutually Exclusive? | |

| |Review |Review packet (not included here) |

|11 | | |

| |Test | |

|12 | | |

Name _________________________

Day 2: Worksheet A : Permutations and Factorials

For all problems: Show your method!

1. How many ways can 7 students line up for lunch?

2. How many ways can 5 paintings be line up on a wall?

3. How many ways can 6 children stand in a line if

there are 10 children to choose from?

4. How many ways can 3 different flags by hung in a

row if there are 8 flags to choose from?

5. PA license plates have 3 letters followed by 4 numbers.

(There are 10 digits and 26 letters available.)

a. If the same letter or number can be repeated, how many can be made?

b. If the same letter or number CANNOT be repeated, how many can be made?

6. How many 4-digit numbers (with place value) can be formed (using 0 – 9)?

7. How many 4-digit numbers (with place value) can be formed if each one uses all the

digits 0, 1, 2, 3, 4 without repetition?

8. In how many ways can 6 bicycles be parked in a row?

Show all work for the following:

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. [pic] (Watch the order of operations here!!)

14. Use the permutation formula and show work for each of the following:

Check your answer on your calculator.

a. [pic] b. [pic]

c. [pic] d. [pic]

Name _________________________

Day 3: Worksheet B : Special Permutations

Show your set up for each problem (not just an answer!)

1. Find the number of ways 10 cheerleaders can make a circular formation.

2. How many ways can 7 children sit in a circle for story time?

3. Find the number of permutations of the letters of these words:

(Remember there are repeated letters!)

a. DEED

b. COMMITTEE

c. CINCINNATI

4. In how many ways can 4 blue, 3 red, and 2 green flags be arranged on a pole?

Mixed Permutation Problems:

You may have to review your notes from the past few days.

5. How many 5-number license plates can be made using the digits 1, 2, 3, 4, 5, 6, 7

d. if repetition of digits is allowed?

e. if repetition of digits is NOT allowed?

6. A teacher wants to write a 4-question test from a total of 12 possible questions.

How many different arrangements of the test can the teacher write?

7. How many ways can 6 people stand in line at the grocery store if Patty and her

son, Ben are together in the front of the line?

8. How many 7-digit telephone numbers are possible if the digits 0 and 1 are not

allowed at the beginning of the phone number?

9. How many seating charts are possible if 25 seats are available and 15 students

need seats? (HINT: Try using a formula instead of writing this one out.)

10. How many ways can we arrange 10 books on a shelf, if we only have room for 8

at a time?

11. Simplify: Show work 12. Use the formula and show work:

[pic] [pic]

Name _________________________

Day 4: Worksheet C : Combinations

Use the combinations formula to simplify each problem. Show the formula!

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

Set up the following problems, then you may use your calculator.

7. How many different 12-member juries be chosen from a pool of 25 people?

8. A test consists of 20 questions, but you are told to answer only 15. In how many

different ways can you choose the 15 questions?

9. How many ways can nine starting players be chosen from a softball team of 15?

10. Four seniors will speak at graduation. If 30 students audition to speak, how many

different groups of 4 speakers can be selected?

11. On a test, a student must answer 6 of the first 8 questions, and 4 of the next 7

questions. In how many ways can this be done?

12. A committee of 4 boys and 3 girls must be chosen from a group of 25 boys and 15

girls. How many ways can this be done?

13. Eight friends are going to a school dance. How many ways can 4 of the friends fit

in Suzie’s car if Suzie is driving? (Therefore she is one of the 4 in her car!)

14. A group of 4 teachers are going to a conference. If there are 12 teachers available,

and Pat must be one of the 12 going, how many ways can the 4 teachers be selected

to go?

Name _________________________

Day 5: Worksheet D : Review Counting Principles

Decide if the problem is an example of a permutation or combination. Set up each problem and find the solution.

1. How many groups of 4 horses would be made if there were 9 horses in the stable?

2. Mike has nine baseball trophies to arrange on the shelf. How many different ways can they be arranged?

3. In math class, there are 24 students. The teacher picks 4 students to help do a demonstration. How many different groups of 4 could she have chosen?

4. In how many ways can 10 people wait in line for concert tickets if Pete is last in line?

5. The teacher has listed 10 short stories and 6 books on the course syllabus. You must pick 4 short stories and 3 books. How many different ways can you do this?

6. How many 3 –digit numbers with place value can be made from the digits 0, 1, 2, 3, and 4 if repetition is not allowed?

7. How many distinct arrangements can be made from the letters a, a, a, b, b, c?

8. How many version of a test (each order is a new version) can a teacher make if she has 20 questions available, but uses only 10 of them?

Name _________________________

Day 7: Worksheet F : Intro to Probability and Simulations

Note: Reduce all Fractions!!

One of these names is to be drawn from a hat. Determine each probability below:

Mary Jenny Bob Marilyn Bill Jack Jerry Tina Connie Joe

1. P(3-letter name) = ____________ 2. P(4-letter name) = ___________

3. P(name starting with B) = __________ 4. P(name starting with T) = __________

5. P(7-letter name) = ______________ 6. P(name starting with S) = __________

7. P(name ending with Y) = ___________

One of these cards will be drawn without looking. Find

8. P( pulling the number 2) = ____________

9. P(5) = ________ 10. P(J) = _________ 11. P(a number) = _________

12. P(4) = ________ 13. P(T) = _________ 14. P(a letter) = __________

One card is drawn from a well-shuffled deck of 52 cards. Find

15. P(ace) = ________ 16. P(face card – K, J, Q) = _________

17. P(a red 10) = ________ 18, P(NOT a diamond) = ______________

A spinner, numbered 1–8, is spun once. What is the probability of spinning…

Name _________________________

Day 8: Worksheet G : Probability of Mutually Exclusive

Events and the Addition Rule

1. Suppose that an event A has probability of [pic]. What is P(A’)? ______________

2. Suppose that the probability of snow is 0.58, What is the probability that it will

NOT snow? ________________

A card is chosen from a well-shuffled deck of 52 cards.

What is the probability that the card will be:

3. a king OR a queen? ______________________

4. a red jack OR a black king? _______________________

5. a face card OR a card with a prime number? ______________________

6. an even card OR a red card? ____________________

7. a spade or a jack? ________________________

A spinner number 1-10 is spun. Each number is equally likely to be spun.

What is the probability of spinning:

8. a 5 or a 7? ____________________

9. a number less than 5 or greater than 8? ____________________

10. a number greater than 7 or an even number? _____________________

11. a multiple of 3 or an odd number? ________________________

12. a factor of 10 or a factor of 8? __________________________

Rolling 2 dice: Find each probability

(Hint: use the chart from your class notes.)

13. P(sum of 12) _____________________

14. P(sum of 12 or 3) = ___________________

15. P(sum of at least 10) = ______________________

16. P(each die is less than 3) = ______________________

17. P(rolling doubles) = ____________________

18. P(not rolling a sum of 7) = ________________________

Name _________________________

Day 9: Worksheet H : Probability of Independent Events and

Multiplication Rule

1. Bag A contains 9 red marbles and 3 green marbles. Bag B contains 9 black marbles and 6 orange marbles. Find the probability of selecting one green marble from bag A and one black marble from bag B.

2. Two seniors, one from each government class are randomly selected to travel to Washington, D.C. Wes is in a class of 18 students and Maureen is in a class of 20 students. Find the probability that both Wes and Maureen will be selected.

3. A box contains 5 purple marbles, 3, green marbles, and 2 orange marbles. Two consecutive draws are made from the box without replacement of the first draw. Find the probability of each event.

a. P(orange first, green second)

b. P(both marbles are purple)

c. P( the first marble is purple, and the second is ANY color EXCEPT purple)

4. If you draw two cards from a standard deck of 52 cards without replacement, find:

a. P(King first, Jack second)

b. P(face card first, ace second)

c. P(2 aces)

MULTIPLE CHOICE:

5. A coin is tossed and a die with numbers 1-6 is rolled. What is P(heads and 3)?

a. 1/12 b. 1/4 c. 1/3 d. 2/3

6. Two cards are selected from a deck of cards numbered 1 – 10. Once a card is

selected, and it is not replaced. What is P(two even numbers)?

a. 1/4 b. 2/9 c. 1/2 d. 1

7. One marble is randomly drawn and then replaced from a jar containing two white

marbles and one black marble. A second marble is drawn.

What is the probability of drawing a white and then a black?

a. 1/3 b. 2/9 c. 3/8 d. 1/6

8. Maria rolls a pair of dice. What is the probability that she obtains a sum that is

either a multiple of 3 OR a multiple of 4?

a. 5/9 b. 7/12 c. 1/36 d. 7/36

Name _________________________

Day 10: Worksheet I : Probability Practice

A jar contains 12 marbles:

4 red

3 blue

2 white

2 green

1 yellow

If one marble is selected, find

1. P(red) 2. P(white)

3. P(red or white) 4. P(not green)

5. P(red, white or blue) 6. P(neither yellow not blue)

Now, suppose 2 marbles are drawn with replacement. Find

7. P(red, then red again) 8. P(yellow, then blue)

9. P(neither marble is white) 10. P(both marbles are green)

Using the same marble jar, now you select 2 marbles without replacement. Find

11. P(red, red) 12. P(yellow, then green)

13. P(No blue) 14. P(red, then white)

15. P(one yellow and one green – in either order!)

16. P(one red and one blue – in either order!)

Challenge: BONUS point for this one!

P(the 2 marbles are different colors)

-----------------------

2

10

9

S

J

7

4

M

5

4

J

10

[pic]

3

2

1

19. an EVEN number? _________ 20. a multiple of 3? ___________

21. a PRIME number? _________ 22. 9? ____________

(note: 1 is not prime.)

8

5

4

6

7

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