Mr.DeMeo - HOMEWORK



Unit 10

Probability

Grade 7

Name:______________________

Teacher:____________________

Period:______________________

Simple Events/Theoretical Classwork Day 1

Important Vocabulary:

Probability: The chance that some event will happen; the ratio of ways a specific event can happen to the total number of outcomes.

[pic]

Relative Frequency- the ratio of the number of observations in a statistical category to the total number of observations.

*Probability can be expressed as a fraction, a decimal, or a percentage.*

Complementary Events: the set of all outcomes in the sample space that are not included in the event. Example: Rolling a 3 on a number cube is 1/6 the complement is 5/6 ( numbers 1, 2, 4, 5, 6)

P(event) + P(complement) = 1

Examples:

1. A fair coin is flipped, what is the probability of getting a ‘tails’?

Percent___________ Fraction___________ Decimal__________

2. The probability that it rains today is 60%. What is the probability that it does not rain?

Percent___________ Fraction___________ Decimal__________

3. A spinner consists of six equal sections numbered 1-6.

What is the probability of the spinner landing on 5?

a) Find P(3). b) What is the probability of getting an even number?

c) What is P(3 or 4)? d) What is the probability of the spinner landing on 7?

4. John has eight red marbles and four blue marbles in a jar. What is the probability that John picks a marble at

random, and it is not red?

Answer the following questions to demonstrate knowledge of probability:

5. What is the sum of the probabilities of all the outcomes in a sample space?

6. The probability of a certain event occurring is [pic] .

Express this probability as a decimal. Express this probability as a percentage.

What is the probability that this event does not occur?

Simple Events/Theoretical Classwork Day 1

7. Which of these cannot be considered a probability of an outcome? Explain.

[a] [pic] [b] -0.59 [c] 1 [d] [pic] [e] 0

[f] [pic] [g] 0.80 [h] 1.45 [i] 112% [j] 100%

[pic] [pic] [pic]

Describe each event as impossible, likely, unlikely, or certain.

8. The probability of tossing a number cube and getting 5 is [pic]. _________________

9. The probability of spinning blue on a spinner is 0. ___________________________

10. The probability of selecting a red marble from a bag of marbles is 0.47. ____________________

11. The probability of selecting a tile with a vowel on it from a box of tiles is [pic]. ________________

12. If a fair die is rolled one time, find the probability of the following outcomes:

[a] rolling a four

[b] rolling an even number

[c] rolling a number greater than four

[d] rolling a number less than seven

Which was most likely to occur (a, b, c or d)?

13. A box contains 5 green pens, 3 blue pens, 8 black pens, and 4 red pens. One pen is picked at random.

[a] What is the probability the pen is green?

[b] What is the probability the pen is blue or red?

[c] What is the probability the pen is gold?

14. The spinner is used for a game. Write each probability as a fraction.

[a] P(3) [b] P(5) [c] P(1 or 2) [d] P(odd) [e] P(a number at most 2)

Simple Events Classwork Day 1

15. A spinner has eight congruent sections, which are colored in the following way:

[i] 2 Red Sections

[ii] 2 Yellow Sections

[iii] 1 Blue Section

[iv] 3 Green Sections

What is the probability of the following outcomes:

[a] Spinning a Red b] Spinning a Red or a Green [d] Not Spinning a Color in the American Flag

*Example 16 Below are three different spinners. If you pick green for your color, which spinner would give you the best chance to win? Give a reason for your answer.

[pic][pic][pic]

Deck of Playing Cards

I. 52 total cards (four suits of each - ♥ ♦ ♣ ♠)

a. Face Cards (Jack, Queen, King)

b. Other cards (2-10, and Ace)

18. If a magician asks you to select one card from a fair deck of cards, find:

[a] P(ace) [b] P(red) [c] P(not a diamond) [d] P(Queen of spades)

[e] Probability of selecting a Spade or a Diamond [f]Probability of selecting a red picture card

[g] Probability of selecting a 1

19. A weather forecast states that there is an 80% probability of rain tomorrow. Which term best describes the likelihood of rain tomorrow?

A. Impossible B. Unlikely C. Likely D. Certain

Simple Events Homework Day 1(2 pages)

1. A spinner with six equal sections is used for a game. The sections are numbered 1-6 Write each probability as a fraction.

P(3) b. P(7) c. P(3 or 4) d. P(even) e. P(not 5)

2. A bag contains 4 red marbles, 3 orange marbles, 7 green marbles, and 6 blue marbles. Express each probability as a fraction:

P(red) b. P(green) c. P(red or blue) d. P(not green) e. P(purple)

3. If the probability that it will snow tomorrow is 0.85, what is the probability that it will not snow tomorrow?

4. There is a 30% chance that it will rain on Saturday. What is the probability that it will not rain?

#’s 5 – 6 Describe each event as impossible, likely, equal chance, unlikely, or certain.

5. The probability of a spinner landing on a shaded section is 53%.

6. The probability of tossing a number cube and rolling a number greater than 1 is [pic].

7. A company that manufactures light bulbs finds that one out of every twenty light bulbs are defective.

a) Express, as a fraction, the probability that a random light bulb is defective. (defective-broken)

b) Express, as a fraction, the probability that a random light bulb is not defective.

c) In a sample of 100 light bulbs, how many bulbs should the company expect to be defective?

d) The manager of your branch of the company tells you that 20% of the light bulbs manufactured are

defective. Is this an accurate statement?

16. There are 4 aces and 4 kings in a standard deck of 52 cards. You pick one card at random. What is the probability of selecting an ace or a king? Explain your reasoning.

______________________________________________________________________________________________

______________________________________________________________________________________________

17. Reasoning: A box contains 150 black pens and 50 red pens. Chris said the sum of the probability that a randomly selected pen will not be black and the probability that the pen will not be red is 1. Explain whether you agree. ______________________________________________________________________________________________

_______________________________________________________________________________Next Page

Homework Day 1

[pic]

Example 18 Decide where each event would be located on the scale above. Place the letter for each event on the appropriate place on the probability scale.

Event:

A. You will see a live dinosaur on the way home from school today.

B. A solid rock dropped in the water will sink.

C. A round disk with one side red and the other side yellow will land yellow side up when flipped.

D. A spinner with four equal parts numbered 1-4 will land on the 4 on the next spin.

E. Your name will be drawn when a name is selected randomly from a bag containing the names of all of the students in your class.

F. A red cube will be drawn when a cube is selected from a bag that has five blue cubes and five red cubes.

G. The temperature outside tomorrow will be -250 degrees.

Example 19. Design a spinner so that the probability of green is 1.

Example 20 Design a spinner so that the probability of green is 0.

Example 21 Design a spinner with two outcomes in which it is equally likely to land on the red and green parts.

______________________________________________________________________________________________

Mixed Review

22. Simplify: 3(2x – 3) – 10(x – 2) 23. -2x and 2x are additive inverses because….

24. Solve: 3x – 5x = 4 25. Solve: 3x > -9 26. Solve: -3x > -9

Outcomes Classwork Day 2

Important Vocabulary:

Outcomes:________________________________________________________________________________

Sample Space: _____________________________________________________________________________

Fundamental Counting Principle:_______________________________________________________________

Example: Complete the tree diagram for tossing a coin three times.

a) P(HHH) = b) P(TTT) =

c) P(at least one H) = d) P(exactly 2 T’s) =

e) If you tossed 4 coins, how many possible outcomes would there be?

Tree Diagrams: Displays all outcomes in detail

Make a tree diagram to represent the sample space of flipping a balanced coin and rolling a fair die.

Total Outcomes:_______

What is the probability of the coin landing on tails and rolling an even number?

A pizza shop offers the following options for a slice of pizza: 1. TYPE: Regular or Sicilian

2. CRUST: Thin or Thick 3. TOPPINGS: Pepperoni, Sausage, Meatball, or Anchovies

Make a tree diagram to represent the sample space of the various slices that could be made.

Outcomes Classwork Day 2

Fundamental Counting Principle (FCP): Allows us to determine the number of outcomes in a sample space by multiplying the number of ways each event can occur.

Examples:

A pizza shop offers the following options for a slice of pizza:

TYPE: Regular or Sicilian

CRUST: Thin or Thick

TOPPINGS: Pepperoni, Sausage, Meatball, or Anchovies

Use the FCP to determine the total number of possible slices of pizza.

1. A restaurant has four different appetizers, three different entrees, and two different desserts on their price-fixed menu. How many different outcomes can there possibly be?

2. If Mr. DeMeo has fifteen pairs of pants, twenty-three collared-shirts, and sixty-four ties; what are the total number of outfits that he can possibly create?

3. If a student rolls two dice, what is the number of total outcomes?

4. Find the total number of different outfits that can be made from the following:

3 different sweaters, 4 turtlenecks, and 2 pairs of jeans.

5. When rolling a fair die and flipping a balanced coin, what is the total possible outcomes?

Example 6 (M5L6) Two friends meet at a grocery store and remark that a neighboring family just welcomed their second child. It turns out that both children in this family are girls, and they are not twins. One of the friends is curious about what the chances are of having 2 girls in a family's first 2 births. Suppose that for each birth the probability of a “boy” birth is 0.5 and the probability of a “girl” birth is also 0.5.

Draw a tree diagram demonstrating the four possible birth outcomes for a family with 2 children (no twins). Use the symbol “B” for the outcome of “boy” and “G” for the outcome of “girl.” Consider the first birth to be the “first stage.”

What is the probability of a family having 2 girls in this situation? Is that greater than or less than the probability of having exactly 1 girl in 2 births?

Outcomes Homework Day 2

1. Create a tree diagram and list the sample space representing all possible outcomes of flipping a coin twice.

(Complete the tree and list the probabilities)

2. Create a tree diagram and list the sample space representing all possible outcomes of rolling a fair die twice.

3. Create a tree diagram and list the sample space representing all possible outcomes of choosing a hat that comes in black, red, or white AND medium or large.

4. Create a tree diagram and list the sample space representing all possible outcomes of choosing peach or vanilla yogurt topped with peanuts, chocolate, strawberries, or granola.

5. At a wedding you can choose from 4 different meats (lobster, steak, chicken, or pork). You can choose from 2 side dishes (pasta or vegetables) and from 2 desserts (fruit or ice cream). How many total outcomes are possible? Use the FCP.

6. At dinner you have the choice of 3 different soups, 4 appetizers, 5 main meals, and 3 desserts. Find the number of possible outcomes of choosing 1 of each course from the menu.

Independent Events Classwork Day 3

Important Vocabulary:

Compound Event: _________________________________________________________________________

Independent Event: ________________________________________________________________________

Dependent Event: _________________________________________________________________________

Each fraction is the theoretical probability of an event.

When all the possible outcomes of an experiment are equally likely, the probability of each outcome is

[pic]

Determine if each of the following events are considered independent or dependent:

[a] Tossing a coin and drawing a card from a deck.

[b] Drawing a marble from a jar, not replacing it, and then drawing a second marble.

[c] Driving on ice and having an accident.

[d] Having a large shoe size and having a high IQ

[e] Not studying for a test and receiving a low test score.

[f] Picking a card from a deck, replacing it, and choosing another card.

[g] Picking a card from a deck, and then choosing another card without replacing the first.

[h] Picking a marble from a jar, replacing it and picking another marble.

[i] Committing a crime and getting arrested.

To find the probability of compound independent events, multiply the probability of each event.

[pic]

Examples 1:

When flipping a coin twice, what is the probability of getting two tails?

Example 2: A game calls for the player to flip a coin and then roll a fair die. Find each probability:

[a] P(tails and 4) [b] P(heads and odd) [c] P(tails and 7)

Independent Events Classwork Day 3

Practice:

1. A person draws a card from a deck of cards, puts the card back and picks again. Find the following probabilities:

[a] P(red and red) [b] P(5 of clubs and 7 of spades)

[c] P(two face cards) [d] P(two spades)

2. There are 4 green marbles, 5 red marbles, 9 blue marbles, and 2 orange marbles in a jar. One marble is selected at random, replaced, and another is selected. Find the following probabilities.

[a] P(green and blue) [b] P(red and orange) [c] P(red and yellow)

[d] P(two blue marbles) [e] P(no red marbles) [f] P(red or blue, and green)

3) An arrangement of 8 students is shown. The numbers of all the students are in a basket. The teacher selects a number and replaces it. Then the teacher selects a second number. Find each probability.

|Row |Student |

|A |1 |2 |3 |4 |

|B |5 |6 |7 |8 |

a) P(student 1, then student 8) =

b) P(student in row A, then student in row B) =

c) P(student in row A, then student 6, 7, or 8) =

Independent Events Homework Day 3

1. A spinner has eight equal sections numbered 1-8. The spinner is spun twice. Find the following probabilities:

[a] P(1 and 2) [b] P(3 and 3) [c] P(odd and even) [d] P(1 and not 1)

[e] P(7 and 0) [f] P(1 and 0) [g] P(not 0 and not 7) [e] P(both numbers < 4)

2. A company produces two different sized light bulbs. One out of every 25 big bulbs is defective. One out of every 50 small bulbs is defective.

a) What is the probability that when purchasing one of each, both will be defective?

b) What is the probability that when purchasing only one small bulb, the bulb will not be defective?

c) In a sample of 200 big bulbs, how many defective bulbs are to be expected?

d) In a sample of 200 small bulbs, how many defective bulbs are to be expected?

3. What is the probability of flipping a coin 3 times and getting heads every time?

4. What is the probability of getting five consecutive tails when flipping a coin five times?

5. A spinner has four equal sections numbered 1 though 4. You spin it twice. Use the sample space below to find each probability. Second Spin

a) P(1,2) b) P(1,odd) c) P(even, odd)

Dependent Events Classwork Day 4

To find the probability of compound dependent events, multiply the probability of the first event and the probability of the second event after the first event happens. (Remember- “Probability Land”- you get to pick one at a time but you get what you want()

[pic]

Describe in your own words the phrase “without replacement”.

____________________________________________________________________________________

Example:

There are 4 green marbles, 5 red marbles, 9 blue marbles, and 2 orange marbles in a jar. One marble is selected at random, and then another is selected without replacement.

a) Find the probability that two blue marbles will be selected

Step 1 : Find the probability of the first event happening:

P(first marble is blue) =

Step 2: Find the probability of the second event happening, assuming the first event did happen:

P(second marble is blue) =

Step 3: Multiply the probabilities of each event:

P(two blue marbles) =

b) Find the probability that the first marble will be red and the second will be green:

P(Red and then Green) =

1. A mason jar contains eighteen marbles in the following colors:

[i] 6 green marbles

[ii] 4 blue marbles

[iii] 7 red marbles

[iv] 1 black marble

What is the probability of the following outcomes without replacement?

[a] P(green and then blue) [b] P(two reds) [c] P(black and then black)

[d] P(two blacks) [e] P(red and then green) [f] P(black and then not black)

[g] P(green and then not red) [h] P(two blues)

Dependent Events Classwork Day 4

2. Five girls and seven boys want to be the two broadcasters for a school show. To be fair, a teacher puts their names in a hat and selects two. Find P(girl, then boy).

Make a Plan: The selections of the two names are (dependent or independent) events? Find the probability of selecting girl first. Then find the probability of selecting a boy after selecting a girl.

Carry out the Plan: P(girl first) = P(boy after girl) =

Final answer: P (girl, then boy) =

3. A student writes the numbers (1-9) on index cards, and then places them in a hat. If another student draws two cards without replacing them, what is the probability of:

[a] P(8 and then 5) [b] P(both digits being even)

[c] P(both digits being odd) [d] P(both digits being perfect squares)

[e] P(1 and then 2) [f] P(9 and then a number less than 9)

[g] P(both numbers greater than 5) [h] P(both numbers are prime)

Easy Medium Challenging

|4. |5. A box contains 20 cards numbered 1-20. You |6. The face cards are removed from a standard deck |

| |select a card. Without replacing the first card, you |of 52 cards, and the rest are set aside. Two cards |

| |select a second card. Find each probability. |are drawn at random from the face cards. Once a card |

| | |is selected, it is not replaced. Find each |

|You select the letter A from the group. Without |a) P(1, then 20) = |probability. |

|replacing the A, you select a second letter. Find | |a) P(2 queens) = |

|each probability. |b) P(3, then even) = | |

|a) P(Z) = | |b) (black jack and then red queen) = |

|b) P(grey) = |c) P(even, then 7) = | |

|c) P(consonant) = | |c) P(black jack and then black card) = |

|d) P(vowel) = | | |

Dependent Events Homework Day 4

1. Mr. DeMeo has to select two students from class to join the SLAM. He decides to choose randomly from a class of eleven girls and nine boys.

[a] What is the probability that he will choose a girl first and then a boy second?

[b] What is the probability he will choose a boy first and then a girl second?

2. There were 5 cards in a bag labeled 0 through 4. Find each probability if two cards are picked with no replacement. (Write the numbers down to help you()

[a] P(2 and then 4) [b] P(2 and then 2) [c] P(1 and then 2 and then 3)

[d] P(prime # and then 0) [e] P(three 0’s) [f] P(# less than 2 and then a 4)

3. In a standard deck of cards: (There are 52 cards in a deck) (4 of each kind) (13 of each suite: ♥♦♣♠)

[a] What is the probability of picking a king or a queen?

[b] What is the probability of picking a king and then a queen with replacement?

[c] What is the probability of picking a king and then a queen without replacement?

[d] What is the probability of picking four consecutive aces without replacement?

Review

4. In a board game, you randomly select one number card and one category card. The possible numbers are 1,2 and 3. The possible categories are Science, History, Sports, Language, and Math. Assume that each outcome is equally likely. Make a tree diagram and sample space to display the outcomes. (*Separate Paper please-This may be collected()

5. William can spend no more than $15 at a carnival. The entrance fee to the carnival is $7, and rides cost $2 each. Which inequality best represents the number of rides r that William can afford?

a) r ≤ 4 b) r < 4 c) r ≤ 11 d) r < 11

Experimental Probability Classwork Day 5

Important Vocabulary:

Theoretical Probability:____________________________________________________________________

Empirical (Experimental) Probability:_____________________________________________________________

When you were spinning the spinner and recording the outcomes, you were performing a chance experiment. You can use the results from a chance experiment to estimate the probability of an event.

THEORETICAL EXAMPLE: What should happen(.

1. A fair coin is flipped four times.

[a] P( first flip will be heads) [b] P( all four flips will be tails)

[c] If you were to flip the coin a total of 100 times, how many times would you expect heads to appear?

EMPIRICAL(EXPERIMENTAL)EX.: BASED ON OBSERVED DATA-What actually did happen(.

Class Activity: Example 2: The experiment requires a brown paper bag that contains 10 yellow, 10 green, 10 red, and 10 blue cubes. The cubes are identical except for their color. Your teacher will conduct a chance experiment. Twenty cubes are drawn at random and replaced. After each cube is drawn, have students record the outcome in the table.

Theoretically, what should happen? A) P(yellow) B) P(green) C) P(red) D P(blue)

|Trial |Outcome |

|1 | |

|2 | |

|3 | |

|4 | |

|5 | |

|6 | |

|7 | |

|8 | |

|9 | |

|10 | |

|Trial |Outcome |

|[pic] | |

|12 | |

|13 | |

|14 | |

|15 | |

|16 | |

|17 | |

|18 | |

|19 | |

|20 | |

1. Based on the 20 trials, estimate for the probability of

a. choosing a yellow cube. b. choosing a green cube. c. choosing a red cube. d. choosing a blue cube.

2. If there are 40 cubes in the bag, how many cubes of each color are in the bag? Explain.

3. If your teacher were to randomly draw another 20 cubes one at a time and with replacement from the bag, would you see exactly the same results? Explain.

3. A probability experiment is conducted. In the experiment, a BALANCED coin is flipped 20 times. The results are displayed in the graph below:

| |1 |2 |3 |4 |5 |

|Probability |[pic] |[pic] |[pic] |[pic] |[pic] |

On a given day, find the probability that Rick eats:

a. Two servings of fruit and vegetables.

b. More than two servings of fruit and vegetables.

c. At least two servings of fruit and vegetables.

d. Find the probability that Rick does not eat exactly two servings of fruit and vegetables.

The diagram below shows a spinner designed like the face of a clock. The sectors of the spinner are colored red (R), blue (B), green (G), and yellow (Y).

[pic]

Spin the pointer, and award the player a prize according to the color on which the pointer stops.

Writing your answers as fractions in lowest terms, find the probability that the pointer stops on:

a. red: b. blue: c. green: d. yellow:

Experimental Probability Classwork Day 5

Practice:

1. A probability experiment is conducted to find the experimental probability of getting various sums when two number cubes are rolled. The results of 50 rolls are shown below:

[a] According to the experiment, what is the experimental probability of rolling a sum of 9?

[b] What is the experimental probability of rolling a sum of 8?

[c] What is the experimental probability of rolling a sum that is greater than 7?

[d] What is the experimental probability of rolling a sum that is greater than or equal to 7?

[e] Which sum is most likely to appear based on the experiment?

[f] Which sums are least likely to appear?

[g] In this experiment, two number cubes were rolled. What is the theoretical probability of getting two 3s? Is this the only way to get a sum of 6?

[h] Why is it that certain sums are more likely to appear than others?

[i] Why is it impossible to roll a 1?

Experimental Probability homework Day 5 (1 of 2 pages)

1. When tossing a coin, what is the theoretical probability of:

a) P (heads) b) P(tails) c) P(heads or tails)

2. Perform your own experiment (get a coin(). Flip a coin 50 times. Record the results.

Tally Total

|Heads | | |

| | | |

|Tails | | |

| | | |

Use your data to find the experimental probability of:

a) P (heads) b) P(tails) c) P(heads or tails)

3. Write a conclusion comparing the results from your experiment and the theoretical probability.

4. How many heads would you expect when flipping a fair balanced coin fifty times?

5. How many primes would you expect when rolling a fair die one hundred times?

6. How many times would you expect to pick a diamond, if you selected a card from a fair deck thirty-two times?

7. How many times would you expect to roll a 5 when rolling a fair die twelve times?

8. The odds of a particular team to win the Super Bowl are 1/8. If these odds stayed consistent every year, how many super bowl titles would you expect this team to have in the next 80 years?

9. A fair die is rolled twice.

How many possible outcomes are there?

What is the probability of rolling a 3 and then a 5? GO TO NEXT PAGE

Experimental Probability homework Day 5

10. A company that produces car parts tests a sample of fifty parts. After testing all fifty parts, they find that 7 parts are defective.

[a] What is the experimental probability of a part being defective?

[b] What is the experimental probability of a part being functional?

[c] How many defective parts would you expect in a batch of 1000 parts?

[d] How could the company find a more accurate representation of their defective parts?

11. A particular game of chance is played by flipping a coin, rolling a fair die, and then picking a card from a fair deck. What is the probability of winning the game if:

[a]Winning means (heads, one, ace) [b] Winning means (tails, odd, black)

Example 12 A seventh grade student surveyed[pic]students at her school. She asked them to name their favorite pet. Below is a bar graph showing the results of the survey.

[pic]

Now suppose a student will be randomly selected and asked what his or her favorite pet is.

a. What is your estimate for the probability of that student saying that a dog is their favorite pet?

b. What is your estimate for the probability of that student saying that a gerbil is their favorite pet?

c. What is your estimate for the probability of that student saying that a frog is their favorite pet?

13. Which of the following shows a proportional relationship?

|Y |13 |12 |9 |

|X |5 |4 |3 |

|Y |15 |12 |9 |

|X |5 |4 |3 |

a) y= x + 3 b) y = 3x c) d)

14. Mark has a total of 600 XBOX games. Of those games 1/3 is violent, out of the violent games 30% use bad language, and out of those games (violent and bad language), 3/5 have are extremely inappropriate. How many games were considered extremely inappropriate?

Theoretical Predictions CLASSWORK Day 6

EXPERIMENTAL PROBABILITY: Determined by OBSERVING and COUNTING outcomes from a sample. This is what ACTUALLY happens!

THEORETICAL PROBABILITY: Determined by what we EXPECT will happen.

Relative Frequency- the ratio of the number of observations in a statistical category to the total number of observations.

The more data collected, the closer the estimates are likely to be to the actual probabilities.

Guided Example:

How many times would you EXPECT to get a B if you spun the spinner to the right 4 times?

To get a B – your chances are[pic]. So multiply the 4 times by [pic] to get your answer of 1.

1. How many times would you EXPECT to get a C if you spun the spinner to the right:

a) 4 times b) 100 times c) 200 times d) 1,000 times

2. How many times would you EXPECT to get an A,B, or C if you spun the above spinner:

a) 4 times b) 52 times c) 64 times d) 100 times

Which letter will the spinner most likely land on? _______ Explain _____________________________

3. How many times would you EXPECT to get a 5 if you rolled the die:

a) 6 times b) 36 times c) 132 times d) 6,000 times

5) A company that produces car parts tests a sample of fifty parts. After testing all fifty parts, they find that 7 parts are defective.

a) What is the experimental probability of a part being defective?

b) How many defective parts would you expect in a batch of 1000 parts?

c) How could the company find a more accurate representation of their defective parts?

6) A school has 1,060 students. The results of a survey are shown.

|Students Surveyed |Students Who Produced Computer Art |

|40 |24 |

If the trend in the table continues, which is the best prediction of the total number of students who produced computer art?

A) 260 students B) 480 students C) 640 students D) 790 students

7) The quality control engineer of Top Notch Tool Company finds flaws in 8 of 60 wrenches examined. Predict the number of flawed wrenches in a batch of 2,400.

Theoretical Predictions CLASSWORK Day 6

Example 8 Which of the following graphs would NOT represent the relative frequencies of heads when tossing 1 penny? Explain your answer. ________________________________________________________________

Part B Jerry indicated that after tossing a penny 30 times, the relative frequency of heads was 0.47 (to the nearest hundredth). He indicated that after 31 times, the relative frequency of heads was 0.55. Are Jerry’s summaries correct? Why or why not?

Part C Jerry observed 5 heads in 100 tosses of his coin. Do you think this was a fair coin? Why or why not?

Jerry and Michael played a game and you need to pick a Blue to win(. The following results are from their research using the same two bags:

Jerry’s research: Michael’s research:

| |Number of Red chips |Number of Blue chips picked | | |Number of Red chips |Number of Blue chips picked |

| |picked | | | |picked | |

|Bag A |2 |8 | |Bag A |28 |12 |

|Bag B |3 |7 | |Bag B |22 |18 |

1. If all you knew about the bags were the results of Jerry’s research, which bag would you select for the game? Explain your answer. Using only Jerry’s research, the greater relative frequency of picking a blue chip would be____________________________________________________________________________________________________

_____________________________________________________________________________________

2. If all you knew about the bags were the results of Michael’s research, which bag would you select for the game? Explain your answer. Using Michael’s research, the greater relative frequency of picking a blue chip would be_____________________________________________________________________________________________________

________________________________________________________________

3. Does Jerry’s research or Michael’s research give you a better indication of the make-up of the blue and red chips in each bag? Explain why you selected this research.______________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

Theoretical Predictions HOMEWORK Day 6

1) Find the experimental probability for Seneca Boys Basketball Team. P(loss) =

Wins: 22 Losses: 3

2) A quality control engineer at a factory inspected 300 glow sticks for quality. The engineer found 15 defective glow sticks. What is the experimental probability that a glow stick is defective? How many glow sticks would the engineer expect to find defective out of 900?

3) A quality control inspector finds flaws in 6 of 45 tools examined. If the trend continues, what is the best prediction of the number of defective tools in a batch of 540?

4) The population of Los Angeles, California, throughout the 20th century is shown in the table to the right.

Between which 2 years did the population increase the most?

Answer between ________________ and _______________________

Based on the data in the table, predict the population of Los Angeles in

the year 2020. Justify your prediction. 

Review

5) During hockey practice, Dane blocked 19 out of 30 shots and Matt blocked 17 out of 24 shots. For the first game, the coach wants to choose the goalie with the greater probability of blocking a shot. Which player should he choose? ______________ Explain

6) After tax (8%), a Bose stereo system costs $5,400. Jim, the salesperson makes 5% commission on his sales. How much commission did Jim earn on his sale?

7) A diver’s elevation is decreasing at a rate of 30 feet per minute. If the diver starts at sea level, what will her elevation be after 2.5 minutes?

A. – 75 feet B. – 12 feet C. 12 feet D. 75 feet

Simulations Classwork Day 7

Vocabulary

Simulation – uses devices such as coins, number cubes, and cards to generate outcomes that represent real outcomes.

How can you use technology simulations to estimate probabilities?

You can use a graphing calculator of a computer to generate random numbers and conduct a simulation.

It is often important to know the probabilities of real-life events that may not have known theoretical probabilities. Scientists, engineers, and mathematicians design simulations to answer questions that involve topics such as diseases, water flow, climate changes, or functions of an engine. Results from the simulations are used to estimate probabilities that help researchers understand problems and provide possible solutions to these problems.

There are five steps in the simulation:

*The first is to define the basic outcome of the real experiment, e.g., a birth.

*The second is to choose a device and define which possible outcomes of the device will represent an outcome of the real experiment, (e.g., toss of a coin, head represents boy; roll of a number cube, prime number (P) represents boy; choice of a card, black card represents boy.)

*The third is to define what is meant by a trial in the simulation that represents an outcome in the real experiment

*The fourth is to define what is meant by a success in the performance of a trial (e.g., using a coin, HHT, HTH, and THH represents exactly two boys in a family of three children)

*The fifth step is to perform ������ trials (the more the better), count the number of successes in the ������ trials, and divide the number of successes by ������, which produces the estimate of the probability based on the simulation.

Real World Application

1. A cereal company is having a contest. There are codes for winning prizes in 30% of it cereal boxes. Find an experimental probability that you have to buy exactly 3 boxes of cereal before you find a winning code.

Be sure to: Choose a model.

The probability of finding a winning code is 30% = [pic]

Use the whole numbers from 1to 10. Let three numbers represent buying a box with a winning code.

Winning Code: 1, 2, 3 Non-winning code: 4, 5, 6, 7, 8, 9, 10

*Step 2 Generate random numbers from 1 to 10 until you get one that represents a box with a winning code. Record how many boxes you bought before finding a winning code.

Ex. If five numbers are generated: 9, 6, 7, 8, 1 1 Represents a winning code

|Trial |Numbers Generated |Boxes Bought |

|1 |9, 6, 7, 8, 1 |5 |

|2 |2 |1 |

|3 |10, 4, 8, 1 |4 |

|4 |4, 10, 7, 1 |4 |

|5 |2 |1 |

|6 |4, 3 |2 |

|7 |3 |1 |

|8 |7, 5, 2 |3 |

|9 |8, 5, 4, 8, 10, 3 |6 |

|10 |9, 1 |2 |

*Step 3 Perform multiple trials by repeating step 2

*Step 4 Find the experimental probability

Look at the simulation and see that 1 of 10 trials, you

bought exactly 3 boxes of cereal before finding a winning code.

The experimental probability is [pic] or 10%.

Practice

1. There is a 30% chance that Chris’s county will have a drought during any given year. He performs a simulation to find the experimental probability of a drought (lack of water) in at least 1 of the next 4 years.

Chris’s model involves the whole numbers from 1-10. Complete the description of his model.

Let the numbers 1 to 3 represent ______________ Let numbers 4 to 10 represent ___________________

Perform multiple trials generating ______ random numbers each time.

Look at the results below and complete the tables.

|Trial |Numbers Generated |Drought Years |

|1 |10, 3, 5, 1 | |

|2 |10, 4, 6, 5 | |

|3 |3, 2, 10, 3 | |

|4 |2, 10, 4, 4 | |

|5 |7, 3, 6, 3 | |

According to the simulation, what is the experimental probability that there will be a drought in the county in at least 1 of the next 4 years?

Simulations Day 7 Homework

Looking Back: Try these

Example 1 A group of seventh graders took repeated samples of size 20 from a bag of colored cubes. The dot plot below shows the sampling distribution of the sample proportion of blue cubes in the bag.

[pic]

1. Describe the shape of the distribution.

2. Describe the variability of the distribution.

3. Predict how the dot plot would look differently if the sample sizes had been 40 instead of 20.

Sample of answers to above question:

1. Describe the shape of the distribution.

Mound shaped, centered around ������.������.

2. Describe the variability of the distribution.

The spread of the data is from ������.������������ to ������.������������, with much of the data between ������.������ and ������.������������.

3. Predict how the dot plot would look differently if the sample sizes had been ������������ instead of ������������.

The variability will decrease as the sample size increases. The dot plot will be centered in a similar place but

will be less spread out.

2. John has to complete a research project on the ape

population in Spain. He is trying to estimate the size of

population of apes. He randomly catches 37 apes and

marks them with paint. He releases apes in jungle. The

following year he observes 250 apes and he found

that 10 were marked with the paint that

he used. Find out the best estimate for the size of the ape population?

Name:_________________________________________Date__________________Period____

Probability Review

������ (event) = ������������������������������������ ������������ ������������������������������������������������ ������������������������������������������������������������������ ������������ ������������������ ������������������������������

������������������������������ ������������������������������������ ������������ ������������������������������������������������������������������������

Terms to remember

Probability (A number between and that represents the likelihood that an outcome will occur.)

Probability model (A probability model for a chance experiment specifies the set of possible outcomes of the experiment—the sample space—and the probability associated with each outcome.)

Compound event (An event consisting of more than one outcome from the sample space )

Tree diagram (A diagram consisting of a sequence of nodes and branches. Tree diagrams are sometimes used as a way of representing the outcomes of a chance experiment that consists of a sequence of steps, such as rolling two number cubes, viewed as first rolling one number cube and then rolling the second.)

Random sample (A sample selected in a way that gives every different possible sample of the same size an equal chance of being selected.)

Inference (Using data from a sample to draw conclusions about a population.)

Experimental probability of an event is the ratio of the number of times the event occurs to the total number of trials.

In 1-10: Determine whether the events described are dependent or independent events.

1. Choosing two cards from a deck of cards without replacement.

2. Picking a marble from a jar, replacing it, and picking another one.

3. Rolling a die and flipping a coin.

4. Spinning a spinner twice.

5. Throwing a printer in the garbage, and not being able to print.

6. Choosing two marbles out of a jar, without replacement.

7. Picking a card, replacing it, and picking another card.

8. Studying and getting good grades.

9. Choosing a student from a class of 25, and picking another from the remaining 24 students.

10. Picking a pen from a box, replacing it, and picking another pen.

In 11-22: Use the diagram of a spinner to the right to answer the questions.

11. What is the probability of getting a 1?

12. What is the probability of getting a 2 or a 3?

13. P(odd) 14. P(even) 15. P(odd or even) 16. P(less than 3)

17. How many total outcomes are there for spinning the spinner twice?

18. Is spinning the spinner twice a compound independent event or compound dependent event?

19. P(1 and 1) 20. P(5 and 5)

21. P(odd and even) 22. P(less than 3 and 7)

In 23-29: Use the following information to answer the questions.

A jar contains 5 red marbles, 7 green marbles, 3 blue marbles, and 10 yellow marbles. In these compound event questions, assume there is replacement.

23. P(red) 24. P(red or blue) 25. P(red and then blue) 26. P(Gold)

27. P(Not gold) 28. P(green or blue, and then yellow) 29. P(not red and red)

30-38 A jar contains 5 red marbles, 7 green marbles, 3 blue marbles, and 10 yellow marbles. In these compound event questions, assume there is no replacement.

30. P(both red) 31. P(red and then yellow) 32. P(red and then blue)

33. P(yellow and red) 34. P(green or blue, and then yellow) 35. P(not red and red)

36. P(orange and red) 37. P(both not orange) 38. P(both yellow)

39. The probability that it will rain tomorrow is 0.7. What is the probability that it will not rain?

40. The chance that a certain event does not occur is 40%. What is the probability that this event does occur?

41. How many times would you expect to get heads if you flipped a coin 400 times?

There are 52 cards in a deck, 13 hearts (red), 13 diamonds (red), 13 clubs (black), and 13 spades (black).

42. If two cards are picked at random from a fair deck, without replacement, what is the probability of getting two hearts?

43. If two cards are picked at random from a fair deck, with replacement, what is the probability of getting two hearts?

44. What is the probability of selecting two cards that are less than 8, without replacement?

45. In a probability experiment, a coin is flipped 200 times. The coin landed on heads 73 times.

a. How many times did it land on tails? b. What is the experimental probability of getting heads?

c. What is the experimental probability of getting tails?

d. Based on experimental probability, how many heads would you expect in 400 flips?

e. Based on theoretical probability, how many heads would you expect in 400 flips?

46. Sam rolled a number cube 50 times. A 3 appeared 10 times. How many times would Sam expect a three show up out of the 50 tosses?

47. A coin is tossed 60 times. 27 times head appeared. Find the experimental probability of getting heads.

48. A coin is tossed 60 times. 27 times head appeared. Find the experimental probability of getting heads.

49. Joey works in a factory. The factory makes i-phones. Joey checks if the I-phones are in good working condition. After checking the first 50 phones, he noticed that 8 of them are defective. Out of the 800 produced that day, how many phones can Joey expect to be defective?

50. A T-Shirt company gives the following options for a customized shirt.

color: Red, Green, or Black

size: Small, Medium, or Large

sleeves: Short or Long

Construct a tree diagram, and list the sample space of all possible combinations

51. How many outcomes do you expect from the above experiment using the Fundamental Counting Principle?

52. Solve 3x – 7 = 23 53. Simplify: 3(5x + 3) + 2( 7 + 3x) 54. Evaluate: 5x -2y, if x = 3 and y = -2

55. What is the constant of proportionality of y = 3x? 56. What is the unit rate 5lbs for $15.85?

57. If the chance of snow is 0.01, is it likely to happen? 58. Testing every 7th part is a _______ sample.

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

Impossible

Equal Chance

Certain

[pic]

[pic]

[pic]

1

0

Very

Likely

Somewhat Likely

Certain

0

Impossible

likely

unlikely

Equal Chance

1

[pic]

[pic]

[pic]

2

1

3

4

4 suits

c. Spades (black)

i. 13 total spalikely

unlikely

Equal Chance

1

[pic]

[pic]

[pic]

2

1

3

4

4 suits

Spades (black)

13 total spades ♠ (2-10, J, Q, K, A)

d. Clubs (black)

i. 13 total clubs ♣ (2-10, J, Q, K, A)

e. Hearts (red)

i. 13 total hearts ♥ (2-10, J, Q, K, A)

f. Diamonds (red)

i. 13 total diamonds ♦ (2-10, J, Q, K, A)

Total Outcomes:_______

B (0.5)

G (0.5)

H (0.5) H H (0.5)(0.5) = 0.25

H (0.5)

T (0.5)

H (0.5)

T (0.5)

T ( )

When one is the numerator it means each event has an equal chance.

First Spin

[pic]

L

B

G

A

E

Z

K

O

[pic]

# of rolls

Sum

Use the results from the survey to answer the following questions.

d. How many students answered the survey question?

e. How many students said that a snake was their favorite pet?

A

A

A

B

B

D

C

C

Graph A

Graph B

|Trial |Numbers Generated |Drought Years |

|6 |8, 4, 8, 5 | |

|7 |6, 2, 2, 8 | |

|8 |6, 5, 2, 4 | |

|9 |2, 2, 3, 2 | |

|10 |6, 3, 1, 5 | |

[pic]

[pic]

5

2

6

3

1

4

................
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