FOM2 - UNIT 8 – PACKETFOM2 Unit 8 Notes: DAY 1- Introduction to Probability

The Fundamental Counting Principle allows us to calculate the exact number of ways that two or more events can occur. For example, if you want to make a sandwich and have a choice of 4 different meats (ham, turkey, roast beef and pastrami) and 3 different breads (white, wheat and rye), how many sandwiches could you make? One way to figure the answer is with a tree diagram:

What if 2 cheeses (American and mozzarella) were added to the choices? Would you really want to draw a tree diagram for that? What if 4 condiments (lettuce, tomato, onion and peppers) were added? As you can see, a tree diagram is not the most efficient method. This is where the Fundamental Counting Principle is useful.

The Fundamental Counting Principle

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Two Events If one event occurs in m ways and another event occurs in n ways, then the

number of ways both events can occur is m·n ways. (4 meats and 3 breads can occur is 4∙3 = 12 ways)

Three or more events If more than two events occur, the method is the same. If three events occur in m, n, and r ways, then the number of ways all events can occur is m·n·r ways. (4 meats, 3 breads and 2 cheeses can occur in 4·3·2 = 24 ways)

Example: Each event can occur in the given number of ways. Find the number of ways all of the events can occur.

Event 1: 4 ways Event 2: 9 ways Event 3: 2 ways Event 4: 3 ways

Example: At a restaurant, you have a choice of 8 different entrees, 2 different salads, 12 different drinks and 6 different desserts. How many different dinners consisting of 1 salad, 1 entree, 1 drink and 1 dessert can you choose?

Sample Space: __________________________________________________________

For each of the following situations, list the sample space.

1.Tossing a coin.

2. Rolling a die.

3. Tossing 2 coins.

4. Tossing 3 coins.

5. Rolling 2 die and adding up the total (like you are playing a game).

Use the box of candy to answer the questions below.

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1) How many pieces of candy are there total?

2) What is the probability of selecting a cherry piece?

3) What is the probability of selecting a lemon piece?

4) What is the probability of selecting an orange piece?

5) Which flavor would you have the largest probability of selecting?

6) Which flavor would you have the lowest probability of selecting?

7) Would you be more likely to select a lemon piece or an orange piece?

8) What is the probability of selecting either a cherry piece or a lemon piece?

9) Your friend wants either a cherry piece or an orange piece . If he or she randomly picked, what is the probability of getting what he or she wants?

10) What is the probability of choosing a lemon, cherry, or orange piece?

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Introduction to Probability

1. You will miss a day of school this year.

2. You wi ll get an A in math class.

3. A tossed coin will land "tails”. 4. You will live to be 100 years old.

5. The sun will rise tomorrow. 6. You have type "D" blood.

7. The next President of the U.S. will be over 21 years old.

8. A card drawn from a standard deck will be a club.

9. You will take a space trip to the moon.

10. You will listen to music today.

11. The next baby born in this state will be a girl.

Probability is a number that cannot be less than or bigger than .

In percent form, probability is a number between and .

If an event is as likely to occur as not to occur, what is the probability?

Theoretical Probability that an event A will occur

All possible outcomes

X X XX X X

In this diagram, there are 9 total outcomes and 3 are

in A so P(A) = 39

Example: Slips of paper numbered 1through 12 are placed in a hat. You have an equally likely chance of choosing any of these integers. Find the probability of the given event.

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Assign a probability to each event.

a) an even number is chosen b) a prime number is chosen c) a factor of 48 is chosen

Example: A cookie jar contains 4 chocolate chip cookies, 8 peanut butter cookies, 5 sugar cookies and 3 oatmeal cookies. Find the probability of the given event.

a) P(chocolate chip) b) P(sugar) c) P(not oatmeal)

Theoretical probability is not always convenient to find. In this case, finding the Experimental Probability may be an option. Experimental probability is found by performing an experiment, taking a survey or looking at the history of an event. You find the probability by

P(A) =

ACTIVITY: Let’s do an experiment on our own. Get a partner and two number cubes from your teacher. You will play a game. Player 1 will win when the sum of the two number cubes is 6. Player 2 will win when the sum of the number cubes is 10. No-one wills if the sum of the two number cubes is 1-5, 7, 8, 9, 11, 12. Tally on your sheets the number of wins. Stop after rolling 20 times. Then calculate the probability of each outcome of the experiment in the last column.

Now let’s compare to Theoretical Probability. Look at the sample space for rolling 2 number cubes below. Circle the outcomes that would make Player 1 win. Now square the outcomes that would make Player 2 win. Use this to answer the questions below.

Theoretical Probability of Player 1: Compare to experimental probability:

Theoretical Probability of Player 2: Compare to experimental probability:

Are you surprised by the results? Why 5

Tally Experimental ProbabilityPlayer 1 Wins(rolls a sum of 6)Player 2 Wins(rolls a sum of 10)Neither Wins

Total Number of Rolls: 20

is the theoretical and experimental probability usually different? Should they be close?

Example: In 1998 a survey asked Internet users for their ages. The results are summarized in the table below.

Age Number of UsersUnder 21 163621-40 661741-60 369361-80 491Over80 6

a) Find the experimental probability that a randomly selected internet user is at most 20 years old.

b) Find the experimental probability that a randomly selected internet user is at least 41 years old.

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FOM2 Unit 8 Homework: DAY 1- Introduction to Probability For each of the following, determine the total number of possibilities.

1. Kone King has 4 different flavors of ice cream, 3 different kinds of cones, and 5 different sizes. How many different cones can they make?

2. Pizza Palace has small & medium pizzas. You can order thin, thick, or stuffed crust. In addition, they have the following toppings: pepperoni, sausage, ham, mushrooms, onions, or peppers. How many 1topping pizzas are available?

3. Jill has 4 different jeans, 6 shirts, 2 belts, and 3 scarves. How many different outfits does she have?

4. You are going to set up a stereo system by purchasing separate components. In your price range you find 5 different receivers, 8 different compact disc players, and 12 different speaker systems. If you want one of each of these components, how many different stereo systems are possible?

5. A pizza shop runs a special where you can buy a large pizza with one cheese, one vegetable, and one meat for $9.00. You have a choice of 7 cheeses, 11 vegetables, and 6 meats. Additionally, you have a choice of 3 crusts and 2 sauces. How many different variations of the pizza special are possible?

Find the probability for each of the following. Show your work!

6. You earned an A on 4 of your last 7 quizzes. What is the probability that you will get an A on your next quiz?

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7. You roll a six-sided die whose sides are numbered from 1through 6. What is the probability of each?

a) P(rolling a 5) b) P(rolling an even number) c) P(rolling less than 7)

8. A jar contains 5 red marbles, 3 green marbles, 4 yellow marbles, and 2 blue marbles. Calculate the probability of the following events.

b) P(yellow) b) P(red) c) P(not green)

9. You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the given event.

c) P(an odd number chosen) b) P( a number less than 7 is chosen)

c) P(a perfect square is chosen) d) P(a prime number is chosen)

e) P(a multiple of 3 is chosen) f) P(a factor of 240 is chosen)

FOM2 Unit 8 Notes: DAY 2- Mutually Inclusive and Exclusive Warm-up

1. A jar contains 2 red marbles, 3 blue marbles, and 1green marble. Find the probability of randomly drawing the given type of marble.

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a) P(a blue or a green marble) b) P(a red or a blue marble) c) P(a green or a red marble)

2. The table shows how students in Mr. Wade's class fared on their first driving test. Some took a class to prepare, while others did not.

a) Find the probability that Paige took the class.

b) Find the probability that Elizabeth did not take the class.

3. The number of students who have attended a football game at Central High School is listed below. Find the probability that a student who has attended a game is a junior or a senior.

Class Freshman Sophomore Junior Senior

Attended 48 90 224 254

Notes: Probability of Compound Events

Compound event consists when there are two or more events.

Consider two sets of number: A= whole numbers between 6 and 12 B =whole numbers less than 5.

Draw a Venn Diagram to represent the situation.

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Class No ClassPassed 64 48

Since there are no numbers in the overlapping part of the circles (INTERSECTION) these two events are mutually exclusive. And can be redrawn like this:

When two events are mutually exclusive, the probability of A or B is:P(A or B) = P(A) + P(B)

Now consider two sets of numbers: A= whole numbers less than 10 B =even numbers less than 15.

Draw a Venn Diagram to represent the situation.

Since there are numbers in the overlapping part of the circles these two events are NOT mutually exclusive. When two events are not mutually exclusive, the probability of A or B is:

P(A or B) = P(A) + P(B)- P(A and B)

Example: Find the indicated probability. State whether A and B are mutually exclusive.

a) P(A) = 0.3 b) P(A) = 0.15 P(B) = 0.55 P(B) = 0.28 P(A or B) = 0.85 P(A or B) =

P(A and B) = P(A and B) = 0.08

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Example: You have a standard deck of cards. If you pick 1 card out of the deck, what is the probability it is a 3 or a club?

Example: Again, you have a standard deck of cards. If you pick 1 card out of the deck, what is the probability it is a face card or a heart?

Finding the complement of an event: Think of the compliment as the opposite. The probability of the complement of an event is:

P( A ' )=1−P( A )

Example: Find P(A').

a) P(A) = 0.67

b) P( A )=3

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Example: Two fair die are rolled. Find the probability of the given event.

a) the sum is not 2 b) the sum is less than 4 OR greater than 10

Example: Students choose one area of science for a project. Biology was chosen by 26% of the students and 18% of the students chose botany. What is the probability that a student chosen at random has selected a project in the field of biology or botany.

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FOM2 Unit 8 Homework: DAY 2- Mutually Inclusive and Exclusive

Find the indicated probability. State whether A and B are mutually exclusive .

1. P(A) = 0.4 2. P(A) = 0.25 P(B) = 0.35 P(B) = __________ P(A or B) = 0.5 P(A or B) = 0.70 P(A and B) = __________ P(A and B) = 0 Mutually Exclusive? __________ Mutually Exclusive? __________

3. P(A) = 5% 4. P(A) = 16% P(B) = 29% P(B) = 24% P(A or B) = __________ P(A or B) = 32% P(A and B) = 0% P(A and B) = __________ Mutually Exclusive? __________ Mutually Exclusive? __________

Find P(A').5. P(A) = 0.34 6. P(A) = 0 7. P(A) = 1

Two six-sided number cubes are rolled. Find the probability of the given event. Refer to the diagram for all possible outcomes.

8. The sum is NOT 8.

12. The sum i

9. The sum is greater than or equal to

4. 10. The sum is less than or equal to 10.

11. The sum is neither 3 nor 7. 12. The sum is greater than 2.

FOM2 Unit 8 Notes: DAY 3 – Independent vs Dependent Events

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Notes: Probability of Independent and Dependent Events

Two events are independent if the occurrence of one has no effect on the occurrence of the other.

What are some examples of independent events?

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The probability that two independent events A and B both occur is: P(A and B) =

P(A)·P(B)

Example: Events A and B are independent. Find the indicated probability.

P(A) = 0.20 P(A) = 0.48P(B) =0.15 P(B) = __________P(A and B) = __________ P(A and B) = 0.16

Example: A bag contains 3 red marbles, 7 white marbles, and 5 blue marbles . You draw 3 marbles, replacing each one before drawing the next. What is the probability of drawing a red, then a blue, and then a white marble?

Example: Three colleagues go to a restaurant and order a sandwich. The menu has 3 different turkey sandwiches, 4 different beef sandwiches and 2 different ham sandwiches.

a) Find the probability that the first colleague chooses a turkey sandwich, the second chooses a beef sandwich and the third chooses a ham sandwich.

b) Find the probability that all three colleagues each choose a different turkey sandwich.

Two events are dependent if the occurrence of one affects the occurrence of the other.

What are some examples of dependent events?

The probability that two dependent events A and B both occur is: P(A and B) = P(A)·P(B following A)

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Example: A bag contains 3 red marbles, 7 white marbles, and 5 blue marbles . You draw 3 marbles WITHOUT REPLACEMENT.

o What is the probability of pulling 2 white marbles?

o What is the probability of drawing a red, then a blue, and then a white marble?

Example: If you randomly pull two cards out of a deck without replacement, what is the probability those two cards are ace?

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FOM2 Unit 8 Homework: DAY 3 – Independent vs Dependent Events Events A and B are independent. Find the indicated probability.

1. P{A) = 0.3P(B) = 0.9P(A and B)=

2. P{A)=

P(B) = 0.3 P(A and B)= 0.06

3. P(A) = 0.75P(B)=

P(A and B) = 0.54

4. Three friends are taking an English class that has a summer reading list. Each student is required to read one book from the list, which contains 3 biographies, 10 classics, and 5 historical novels.

a) Find the probability that the first friend chooses a biography, the second friend chooses a classic, and the third friend chooses a historical novel.

b) Find the probability that the three friends each choose a different classic.

Determine whether the events are independent or dependent. Then find the probability.

5. Yana has 4 black socks, 6 blue socks, and 8 white socks in his drawer. If he selects three socks at random with no replacement, what is the probability that he will first select a blue sock, then a black sock, and then another blue sock?

6. Event A is drawing a spade from a deck of cards and event B is drawing a club from the remaining cards.

7. Event A is rolling a 6 on a fair die and event B is rolling an even with a different fair die.

8. A bag contains 5 red skittles, 8 orange skittles, 3 yellow skittles, and 4 green skittles. You pick out 1 skittle but put it back. What is the probability of drawing a yellow, then a green, and then a red skittle?

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Determine whether the events are independent or dependent. Find the probability.

9. A black die and a white die are rolled. What is the probability that a 3 shows on the black die and a 5 shows on the white die?

10. In a board game, three dice are rolled to determine the number of moves for the players. What is the probability that the first die shows a 6, the second die shows a 6, and the third die does not show a 6?

11. A bag contains 12 red marbles, 9 blue marbles, 11yellow marbles, and 8 green marbles. Three marbles are randomly drawn from the bag one at a time and not replaced. Find the probability that red, blue, and green marbles are selected, and in that order.

12. You are playing a game that involves drawing three numbers from a hat. There are 25 pieces of paper numbered 1 to 25 in the hat. Each number is replaced after it is drawn. What is the probability that each number is greater than 20 or less than 4?

13. Nine slips of paper are numbered 1 to 9 and are placed in a bag. You randomly draw two slips. You do not replace the first slip of paper before selecting the second. What is the probability that the first number is odd and the second is even?

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