Probability

Notes

Name _____________________ Color __________

Probability 8.1

Probability is _________________________________________________________________________

Probability Continuum

Probability can be written as a ____________________________________________________________

Example: Probability of getting a tail when tossing a coin is ½ or .5 or 50%

Experiment Example: Tossing a coin is the experiment.

Trial Example: Each time you toss the coin is a trial.

Outcome Example: Heads or Tails are possible outcomes when tossing a coin.

Event Example: Getting a heads when you toss the coin is the event.

Complement of an Event: ___________________________________________________________

Example: Probability of rolling a 4 = 1/6. Not rolling a 4 = 5/6.

Uniform Probability: An event where all the outcomes ___________________________________

Practice

Write impossible, unlikely, equally likely, likely, or certain

______________________________

_______________

_______________

_______________

1. It is ________________ to draw a striped pebble from the bag. _______

2. Drawing a white pebble from the bag is _________________. _______

3. Drawing a spotted pebble from the bag is _______________________. _______

4. If you reach into the bag, it is _______________ that you will draw a pebble. _______

5. You are _______________ to draw a pebble that is not black from the bag. _______

Practice on your own!Determine whether each event is impossible, unlikely, equally likely, likely, or certain.

1. Rolling an even number on a number cube labeled 1 through 6 __________________

2. Picking a card with a vowel on it from a box of cards in which each letter on the alphabet is written on a card. __________________

3. Spinning a number greater than 2 on a spinner with 10 equal sections marked 1 through 10 __________________

4. Drawing a red marble from a bag of black, blue, and green __________________

5. Flipping a coin and getting heads or tails __________________

6. Rolling a number that is less than three 5 times in a row on a number cube labeled 1 through 6 __________________

7. A bag contains 3 green marbles, 7 blue marbles, and 2 black marbles. The Probability of randomly picking a green marble is 1/4 . What is the probability of not picking a green marble?

8. A spinner has 8 equal sections labeled 1 through 8. The probability of spinning a number that is greater than or equal to 6 is 3/8. What is the probability of spinning a number not greater than or equal to 6?

9. The probability of randomly drawing a red card from a bag is 3/10. What is the probability of not drawing a red card?

10. Myra almost always spends at least 45 minutes on the treadmill. If Myra got on the treadmill at 5:20 PM, estimate the probability that she will still be on the treadmill at 6:00.

11. Morris rarely arrives home before 4:00 pm. It is now 3:20 pm. Estimate the probability that Morris will arrive home in the next 30 minutes.

Sample Space

Tree Diagram __________________________________________________________________________

8.3

Example Meal Choices: Burger, Hot dog and Pizza Drinks: Coke or Sprite

Sample Space ________________________________________________________________________

Example ____________________________________________________________________________

1. Andy flips a coin and rolls a dice. a) Draw a tree diagram, b) What is the sample space (i.e. all possible outcomes)? How many outcomes are in the sample space?

2. You are playing a game using the spinner below. The game requires that you spin the spinner twice. For example, one outcome could be Yellow on 1st spin and Red on 2nd spin. List the sample space (all the possible outcomes) for the two spins.

3. List the sample space for the chance experiment of flipping a coin twice.

Counting PrincipleCounting Principle ______________________________________________________________________

8.5

# of ways a certain outcome can occurTotal Possible Outcomes (Sample Space)

Example ______________________________________________________________________________

1) For the lunch special at Nick’s Deli, customers can create their own sandwich by selecting 1 type of bread and 1 type of meat from the selection below.

a) In the space below, list all the possible sandwich combinations using 1 type ofbread and 1 type of mean.

b) If Nick decides to add whole wheat breadas another option, how many possible sandwich combinations will there be?

2) Helen is preparing candy bags for the children at a party. She has 2 flavors of lollipops, 4 types of candy bars, and 6 flavors of chewy candies. If each bag contains one piece of each type of candy, what is the total number of possible candy combinations for the bags?

A) 12 B) 15 C) 36 D) 48

3) Peter has 6 sweaters, 4 pairs of jeans, and 3 pairs of shoes. How many different outfits can Peter make using one sweater, one pair of jeans, and one pair of shoes?

A) 13 B) 36 C) 72 D) 144

4) Erin wants to make a sandwich from the main ingredients shown in the table below.

In the space below, list all the possible ways Erin can make a sandwich using one type of bread and one main ingredient.

Probability =

Below are the results of the last math test. The teacher determines that anyone with a grade of more than 65 passed the test (65 is failing). Give the probability for the indicated grade.

1. P(70) 2. P(100) 3. P(80) 4. P(passing)

5. P(grade> 80) 6. P(60) 7. P(failing) 8. P(grade < 80)

9. Consider an experiment of randomly selecting a letter from the word: number

a. What is the sample space? List the probability of each outcome in the sample space.

b. What is the probability of selecting a vowel?

c. What is the probability of selecting the letter z?

10. A chance experiment consists of spinning the two spinners below.

a. Make a tree diagram and give the total possible outcomes.

b. Find the probability of the event of getting a red on the first spinner and a red on the second spinner.

c. Find the probability of a red on at least one of the spinners.

Class Activity

Complete the table of theoretical probabilities for rolling a number cube.

Predict the number of times each number will be rolled out of 30 total rolls.

Roll a number cube 30 times. Complete the table for the frequency of each number and then find its experimental probability.

Look at the tables you completed. How do the experimental probabilities compare with the theoretical probabilities?

By performing more trials, you tend to get experimental results closer to the theoretical probabilities.We will create a table for the class.

Experimental Probability 8.2

Experimental Probability__________________________________________________________

______________________________________________________________________________

Example 1: During football practice, Sam made 12 out of 15 field goals. What is the probability he will make the field goal on the next attempt?

Example 2: Mrs. Leaderhouse’s students have taken out 85 books from the library. 35 of them were fiction.

What is the probability that the next book checked out will be a fiction book?

Example 3: Emma made 9 out of 15 foul shots during the first 3 quarters of her basketball game. What is the probability that the next time she takes a foul shot she will make it?

Example 4: Christina scored an A on 7 out of 10 tests. What is the probability she will score an A on her next test?

Example 5: After a movie premiere, 99 of the first 130 people surveyed said they liked that movie.

What is the probability that the next person surveyed will say he or she liked the movie?

What is the experimental probability that the next person surveyed will say he or she did not like the movie?

Experimental Probability Practice

The table shows a breakdown by format of total music sales in the United States in 2004.

1. What is the experimental probability that any random music purchase in 2004 was a CD?

2. What is the experimental probability that any random music purchase in 2004 was a digital single?

3. What is the experimental probability that any 4. Which combination of salesrandom music purchase in 2004 was not aMusic has an experimental probability

Video? of 120?

Choose the letter for the best answer.

5. Ethan hits 4 ringers in 10 attempts while 6. Jay beats Terry at table tennispitching horseshoes. What does an experimental 3 out of 5 games. What is the

of 25 describe? Experimental probability that

Terry with win the next game?A P(horseshoes) B P(missed shots

C P(attempts) D P(ringers) A 12 B

25

C 35 D 1

7. Pat counts 10 classmates out of 36 people in the 8. Macy makes 15 of 20 free throwslibrary. What is the experimental probability that at a basketball practice. What is the next person will be a classmate? the experimental probability that

she will miss her next free throw?

A536 B

136 A

14 B

23

C518 D

110 C

12 D

34

Theoretical Probability

Theoretical Probability ___________________________________________________________________

Example 1: Andy has 10 marbles in a bag. 6 are white and 4 are blue. Find the probability as a fraction, decimal, and percent of each of the following:

a) P(blue marble)

b) P(white marble)

Example 2: If there are 12 boys and 13 girls in a class, what is the probability that a girl will be picked to write on the board?

Example 3: There are 8 black chips in a bag of 30 chips. What is the probability of picking a black chip from the bag?

Example 4: There are 2 small, 5 medium, and 3 large dogs in a yard. What is the probability that the first dog to come in the door is small?

Example 5: What is the probability of getting a tail when flipping a coin?

Example 6: What is the probability of rolling a 4 on a dice?

8.2

Theoretical Probability

An experiment consists of rolling two fair number cubes. Write out the sample space below. Find the probability of each event.

1,1 3,1 6,1

1,2 2,2 5,2 6,2

4,3

1,4 4,4

3,5 5,5

2,6 6,6

1. P( total shown = 3) 2. P(total shown = 7) 3. P(total shown = 9)

4. P(total shown = 13) 5. P(total shown >8) 6. P(total shown <7)

7. A bag contains 9 pennies, 8 nickels, and 5 dimes. How many quarters should be added to the

bag so that the probability of drawing a dime is 16?

8. What is the probability of randomly choosing a white counter from a bag of 12 red, 12 white, 12 green, and 12 blue counters.

9. When tossing two fair coins and having one land on tails and one land on heads.

10. Rolling a number greater than 1 on a fair number cube.

11. Randomly drawing an orange disk from a bag of 14 black disks, 4 blue disks, and 12 orange disks.

12. Randomly drawing 1 of the 6 R’s from a bag of 100 Scrabble tiles

Independent Events

Independent _________________________________________________________________

Example: Find the probability of choosing a green marble at random from a bag containing 5 green and 10 white marbles and then flipping a coin and getting tails.

Example: The spinner at the right is spun once and a card is drawn from a deck of 4 cards labeled A, B, C, and D. Find the following probabilities:

a) P(3 and A) _______ b) P(5 and C) ________

c) P(4 and B or C) _______ d) P(not 4 and C) _______

d) P(1 and not D) _______

With Replacement ____________________________________________________________

Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue?

Example: A jar contains 8 marbles: 3 black, 2 yellow, 2 blue, and 1 purple. Two marbles are drawn with replacement. What is the probability?

a) P(black and yellow) ______

b) P(black and then another black) ______

c) P(yellow or purple, then blue) ______

8.7

Dependent Events

Dependent ___________________________________________________________________

Example: A bag contains 9 blue marbles and 3 red marbles. Two marbles are drawn at random.What is the probability of getting one red marble and then one blue marble?

Example: Two students are selected from a class. There are 12 girls and 13 boys. What is the probability that 2 girls are selected?

Example: Randy has 4 pennies, 2 nickels, and 3 dimes in his pocket. If he randomly chooses 2 coins, what is the probability they are both dimes?

Without Replacement __________________________________________________________

Example: A bag contains 3 blue and 5 red marbles. Find the probability of drawing 2 blue marbles in a row without replacing the first marble.

Example: A jar contains 14 marbles: 3 black, 4 yellow, 5 blue, and 2 purple. Two marbles are drawn without replacement. What is the probability?

a) P(black and yellow) ______ b) P(black and blue) ______

c) P(yellow or purple, then blue) ______

Frequency Tables and ProbabilityExample 1

A dice is thrown and number showing is recorded.

How many times was the dice thrown?

Find the estimated (experimental) probability for each of the six numbers:

P(1) P(2) P(3) P(4) P(5) P(6)

What is the estimated probability that the dice will show:

a) An even number b) A number less than 5 c) Not a 2

Example 2

Bag of MarblesColor Number of MarblesRed 30

Yellow 12Green 8Blue 5

a) What is the probability that a person will choose a Red Marble from the bag?

b) After the red marble is chosen and not replaced, what is the probability the next marble chosen is a green or a blue?

c) What is the probability that the next marble chosen is not a yellow marble?

d) What is probability of choosing a green then a blue without replacing the first marble?

Example 3

Based on the survey below, find the following probabilities.

What is the probability that:

a) A seventh grader chooses English as their favorite subject?

b) A student chooses Math / Science as their favorite subject?

c) An 8th grade student chooses History as their favorite subject?

Probability Predictions

1. A potato chip factory rejected 2 out of 9 potatoes in an experiment. If there is a batch of 1200 potatoes going through the machine, how many potatoes are likely to be rejected?

2. Based on Colin’s baseball statistics, the probability that he will pitch a curveball is 1/4. If Colin throws 20 pitches, how many pitches most likely will be curveballs?

3. If John flips a coin 210 times about how many time should he expect the coin to land on heads?

7.1

4. If Jordan makes 15% of all the shots he takes on goal, how many can he expect to make if he takes 60 shots.

5. Based on Sherry’s past performance, the probability that she gets an A on her math test is 80%. If she takes 15 test this year, how many can she expect to get an A on?

6. Cassie rolls a fair number cube with 6 faces labeled 1 through 6. She rolls the number cube 300 times. Which result is most likely?

A) Cassie will roll a 1 or a 2 about 50 times.B) Cassie will roll a 1 or a 2 exactly 50 times.C) Cassie will roll an even number about 150 times.D) Cassie will roll an even number exactly 150 times

7. Jake plays a game of throwing a ball at a target.

The table shows information about the probability of each possible score.Work out the value of x.

Score 0 1 2 3 4 5

Probability 0.09 X 0.18 0.16 0.21 0.30

8. The table shows the midday temperature on each day for ten days.

Day 1 2 3 4 5 6 7 8 9 10

Temperature (C) 13 14 12 10 13 16 14 13 18 16

What is the probability that on the next day, the temperature will be 13C?

9. The two-way table shows some information about how some students travelled to school yesterday.

Walk Bus Cycle Total

Boys 4 4 13

Girls 2 6 3 11

Total 7 10 7 24

a) Work out the probability that this student will be a boy who walked to school

yesterday.________

b) Probability that a girl cycles to school.__________

c) Probability that a student takes a bus to school. ______________

10. The table shows the probabilities that a spinner will land on 1 or 2 or 3 or 4 or 5

Number 1 2 3 4 5

Probability 0.15 0.20 0.10 0.25 0.30

Pete spins the spinner once.

(a) Work out the probability that the spinner will land on a number greater than 2

Elinor is going to spin the spinner 200 times.

(b) Work out an estimate for the number of times the spinner will land on 5

11. Nancy spins the spinner at the right 60 times Predict how many times the spinner will land on the number 2.

12. If the carnival spinner lands on 10, the player gets a large stuffed animal.

Suppose the spinner is spun 30 times. Predict how many large stuffed animals will be given away.

Fair or Unfair.

13. Roll two fair number cubes labeled 1 – 6. Player A wins if both numbers are the same. Player B wins if both numbers are different.

14. Roll two fair number cubes. Add the numbers. Player A wins if the sum is 5 or less. Player B wins if the sum is 9 or more.

Simulations

A simulation is an experiment that is designed to act out a given situation. Simulations often use models to act out events that would be difficult to perform.

Example 1 – The weather forecast states that there is a 75% chance of snow tomorrow. Describe a method you could use to simulate this situation.

75% = 75100 or

34 Write 75% as a fraction in simplest form.

Since there is a 75% chance of snow, there is a 25% or 14 chance that it will not snow. Place 3 red

marbles to represent snow, and 1 blue marble to represent no snow in a bag and randomly pick one marble. Repeat this selection several times to find the experimental probability that it will snow tomorrow.

8.4

Example 2 – A fresh-squeezed juice store offers 5 different types of fruit juice in a small or large size. If each type of juice and size is equally likely to be chosen, describe a model that could be used to simulate the orders of the next 6 customers.

Since there are 5 types of juices and 2 sizes, choose a method that has 10 possible outcomes, such as tossing a coin and spinning a spinner that has 5 equal sections. Let each specific outcome represent a different choice.

Toss the coin and spin the spinner 6 times to simulate the choices made by 6 customers.

Try these

1. The questions on a multiple-choice test each have 3 answer choices. Describe a model that you could use to simulate the outcome of guessing the answers to a 25-question test.

2. Mr. Jenkins needs to choose 3 captains randomly for teams for a game. If there are 15 students in his classroom, describe a model that he could use to simulate choosing these 3 captains.

3. Suppose during springtime it rains about 40% of the time when school is dismissed for the day. Describe a model that could be used to simulate whether it will be raining when school is dismissed on a particular day during springtime.