# Date. Probability. Chapter

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1 Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games of chance are designed so that the customer loses most of the time. For example, the chance of winning a lottery where you pick 6 numbers out of 49 is in almost 4 million! You have a better chance of being struck by lightning.. a) List contests, lotteries, or games that you have entered. b) How often have you won? Probability MHR

2 Skills Practice : Fractions, Decimals, and Percents. a) You have several quarters. Write the amount shown in 2 ways. Use a cents symbol Use a dollar sign b) Write the amount as a fraction of a dollar. Show the fraction in 2 ways. = c) Percent means out of. The amount shown in part b) is % of a dollar. 2. This measuring tape shows foot a) One foot equals inches. b) Half a foot is inches. c) How many inches are in _ 4 foot? Another way of saying one-fourth is one-. d) How many inches are in 3 _ 4 foot? e) Show 6 inches as a percent of foot. f) Express inch as a fraction of foot. MHR Probability

3 3. Without using a calculator, complete the table. Fraction Decimal Percent _ 2 _ 4 _ % 85% _ 3 2 _ % 3% 0 _ 0 0 _ Skills Practice : Fractions, Decimals, and Percents MHR

4 Date. What s the Chance? Focus: theoretical probability, number sense Warm Up. a) How many weeks are in 2. a) How many seasons are in year? year? b) E ach season is the same length. How many weeks a year? are in each season? 3. Add. 4. What fraction of a dollar is each coin? a) = = b) 20% + 25% + 30% + 5% = = c) _ + _ + _ + _ = = Calculating Theoretical Probability. There are 52 cards in a standard deck of cards. There are 4 different suits. Two suits have red symbols. These are the hearts and diamonds. Two suits have black symbols. These are the clubs and spades. Each suit has numbered cards from 2 to 0, plus a jack, a queen, a king, and an ace. A A A A A A A b) How many weeks are in half A heart diamond club spade You have a full deck of cards. What is the probability of picking the following card? a) a heart b) a black card c) a red card d) an ace MHR Probability

6 4. a) What does this roll of a die show? b) What is the probability of rolling a 2 with die? Write your answer as a fraction. c) What is the probability of rolling a 5? Die is the singular form of the word. d) Create a bar graph showing the probability of rolling each number when you roll die. Include a title for the graph. Label each axis. e) What is the probability of rolling a die and getting a 7? f) Explain your answer to part e). 5. You flip a coin. Create and label a circle graph showing the probability of getting heads or tails. Include a title. Label each sector. MHR Probability

7 6. a) State the probability of the spinner below landing on each colour. Write your answer as a percent. yellow blue Blue: red blue Green: red green green green blue blue Red: Yellow: b) What is the probability of the spinner landing on yellow or blue? c) What is the probability of the spinner landing on green or blue? d) What is the probability of the spinner not landing on blue? Check Your Understanding. Fill in each blank with the appropriate phrase. It will happen. It is not likely to happen. It is likely to happen. It will not happen. It might happen, it might not. 00% 5% 99% 50% % 49% 0% What s the Chance? MHR 9

8 .2 In a Perfect World Focus: theoretical probability, experimental probability, number sense Warm Up. Write _ 0 in lowest terms. 40 _ 0 40 = 3. Write 90% as a fraction in lowest terms. 2. Write 3 equivalent fractions for _ 2. _ 2 = 4. What percent of the bar is shaded? 5. Shade 75% of the cylinder. 6. What is the chance of picking a king from a deck of 52 cards? Show your answer as a fraction in lowest terms. Collecting Data to Calculate Probability Imagine flipping this penny 0 times. In a perfect world you would get 5 heads and 5 tails. This is theoretical probability.. Answer the following questions as though you were in a perfect world. a) What would happen if you flipped a coin 50 times? b) If you rolled a die 60 times, how many 3s would you get? c) If you cut a deck of cards 40 times, how many hearts would you get? 2. In a perfect world, the of flipping heads is 50%. 0 MHR Probability

9 Experimental probability is the chance of something happening based on experimental results. After collecting data, it is useful to compare experimental probability with theoretical probability. 3. a) Create and label a bar graph showing the perfect world results for rolling a die 60 times. Title the graph. Go to pages 2 to write the definition for experimental probability in your own words. b) Roll a die exactly 60 times. Record your results in the tally chart. c) Create and label a bar graph showing your results in part b) d) For each of your results, express the experimental probability as a fraction. = 2 = 3 = 4 = 5 = 6 = In a Perfect World MHR

10 4. a) Create and label a bar graph showing the perfect world results for cutting a deck of cards 40 times. Title the graph. Label each axis. b) Record the results for obtaining each of the 4 suits when you cut a deck of cards exactly 40 times. c) Create and label a bar graph showing your results in part b). d) For each of your results, express the experimental probability as a fraction and then a percent. Clubs: or % Spades: or % Hearts: or % Diamonds: or % 2 MHR Probability

11 5. a) Create and label a circle graph showing the perfect world results for flipping a coin 50 times. Include a title. Label each sector. b) Flip a coin exactly 50 times. Record your results in the tally chart. Heads Tails c) Create and label a circle graph for the results obtained in part b). Check Your Understanding. a) Did anyone in the class get perfect world results for all 3 of the experiments? YES NO b) Explain why few, if any, people in the class received perfect world results for all 3 of the experiments In a Perfect World MHR 3

12 Tech Tip: Experimenting with a Random Number Generator You can use a graphing calculator to simulate experimental probability. Follow the instructions to check out several different applications. Using a TI-83+ Graphing Calculator. Press MATH. Scroll right so that PRB is highlighted. 2. Press 5 to select 5:randInt(. This command tells the calculator to generate random integers. 3. a) To simulate flipping coins, enter,2). Make sure there are no spaces between the characters. This tells the calculator to select either the number or the number 2. Mentally assign heads or tails to each number. For example, is heads, 2 is tails. Continue pressing the ENTER key to generate more random tosses. 4 MHR Probability

13 b) To simulate selecting the suit of a card, enter,4). Make sure there are no spaces between the characters. A A A A A A heart diamond club spade A A This tells the calculator to select an integer from to 4. Mentally assign suit to each of the 4 numbers. Continue pressing the ENTER key to generate more random suits. c) To simulate selecting the value of a card, enter, ). d) To simulate selecting the exact card, enter, ). e) To simulate rolling die, enter, ). 4. Press ENTER. A random integer from the acceptable range of values will be displayed. Continue pressing ENTER to generate more random numbers Tech Tip: Experimenting with a Random MHR 5 Number Generator

14 Skills Practice 2: Equivalent Fractions The word equivalent comes from 2 smaller words. equi = equal valent = value Equivalent fractions are fractions that have the same value. The top number of a fraction is called the numerator. The bottom number of a fraction is called the denominator. 3 _ 4 numerator denominator Go to pages 2 to write the definition for equivalent fractions in your own words. Give an example. Example Look at the 3 bars below. Write the fraction of each bar that is shaded In this example, the same amount of each bar is shaded. These visuals show. = =. a) Write the fraction of each bar that is shaded. 5 b) What is an equivalent fraction for 2_ 3? 5 6 MHR Probability

15 2. Write the fraction of each circle that is shaded. a) b) = = c) = 3. a) Write 4 equivalent fractions for _ 2. b) Explain or show how you developed the fourth fraction above. 4. a) Develop 3 visuals to show your own equivalent fractions. b) Write the fractions. 5. Fill in the blanks to create equivalent fractions. _ = 2_ 2 = _ 6 = _ 0 = 20 _ = _ _ = = 250 _ Skills Practice 2: Equivalent Fractions MHR 7

16 .3 Roll the Bones Focus: theoretical probability, experimental probability, number sense Warm Up. Write 3 equivalent fractions for _ Write each fraction in lowest terms. _ 4 2 = 6 _ 8 = 2. Write each fraction as a decimal. _ 4 = _ 5 = 4. There are 5 students in a class. Five are girls. Write the fraction of the class that is girls in lowest terms. 5. The bar graph shows attendance at a movie theatre for week. a) How many people saw the movie on Wednesday? b) How many people saw the movie on Friday? c) How many people saw the movie last week? Number of People Sun Movie Attendance Mon Tues Wed Thurs Fri Day of the Week Sat Rolling Dice. Suppose you roll 2 dice. a) What is the smallest total you can get? b) What is the greatest total you can get? c) How many different totals are possible? d) If you roll a pair of dice 50 times, predict the number of times that the total will be 7. 8 MHR Probability

17 2. a) Roll 2 dice exactly 50 times. Add the 2 numbers showing. Record the number of times each total occurs. Sum of the Dice 2 Tally Total Times Rolled b) Create a bar graph showing your results. Include a title. Title the y-axis, Total. Title the x-axis, Sum of the Dice. Choose an appropriate scale for the y-axis. c) Did you roll each of the sums an equal number of times? YES NO d) Suggest some reasons for your answer Roll the Bones MHR 9

18 There is only way to roll a 2 with 2 dice. You need a on each die. There are 2 ways to roll a 3. You can have a on the first die and a 2 on the other. Or, you can have a 2 on the first die and a on the other. 3. a) Determine all the possible combinations for rolling 2 dice. Example: (, ) Sum of the Dice Possible Combinations Number of Combinations 2 (, ) 3 (, 2) (, ) Total Number of Combinations 20 MHR Probability

19 b) Create a bar graph showing the Sum of the Dice versus Number of Combinations. Include a title. Title the y-axis, Number of Combinations. Title the x-axis, Sum of the Dice. Choose an appropriate scale for the y-axis. c) Which sum has the highest theoretical probability of being rolled? d) Does your answer to part c) match your experimental results? YES NO e) Why do you think this is the case? 4. When you roll 2 dice, list all of the combinations that make a sum of 7 or greater Roll the Bones MHR 2

20 5. a) Complete the table. Write the fractions in lowest terms. Round each percent to the nearest whole number. Tech Tip: Suppose that you made 5 rolls. You rolled 2 twice. Use your calculator to show 2_ 5 as a percent. If your calculator has a % key, enter 2 5 % 2 is 40% of 5. Sum of the Dice Number of Combinations Fraction of the Total Number of Combinations Percent of the Total Number of Combinations 2 _ = 3% Total b) List the pairs of sums that have the same theoretical probability of occurring. c) The likelihood of rolling a total of 3 with 2 dice is the same as the total of the likelihood of rolling 2 other combinations. What are those 2 combinations? and d) As a percent, what is the chance of rolling 2 dice and obtaining a total of 7 or greater? 22 MHR Probability

21 6. a) Add all of the class s results from #2a) and record the data in the appropriate row of the tally column. Calculate the percent of the total for each sum. Sum of the Dice Class Tally Total For Class Results Percent of Total b) Graph the results. Include a title. Title the y-axis, Percent of Total. Title the x-axis, Sum of the Dice. Check Your Understanding. Which graph is closer in shape to the graph in #3? The graph in #2 or the graph in #6? 2. Why do you think this is so? Roll the Bones MHR 23

23 2. a) Flip all 3 coins exactly 40 times. Record your results in the table. Successful (Got 3 heads) Unsuccessful (Did not get 3 heads) Tally Total b) How many successful results did you get? Show this as a fraction of the total sample. c) State the number of successful results as a percent. 3. In the chart below, list or draw all of the possible outcomes for flipping 3 coins at once. First Coin Second Coin Third Coin 4. a) What is the theoretical probability of a successful result? Show your answer as a fraction and a percent. b) What is the theoretical probability of an unsuccessful result? Show your answer as a fraction and a percent Heads, Heads, Heads MHR 25

24 5. a) Record the individual results of the class from #2a) in the table. Add the class results for Successful and Unsuccessful. Successful Unsuccessful Individual Results Total b) How many flips are in this sample? students 40 flips each = flips c) Calculate the overall percent of successful results. d) Create a circle graph showing the results from part a). Estimate the size of each fraction of the circle. Include a title. Label each sector. 26 MHR Probability

26 .5 Free Coffee Focus: experimental probability, simulation Warm Up. The theoretical probability of winning a prize in a lottery is in 5. Write this as a fraction and a percent. 2. The weather report says there is a 70% chance of snow. Write the probability of it snowing as a decimal and a fraction. 3. You roll 2 dice. Circle the probability of rolling a sum of 7. Impossible Very Likely Likely Not Likely Certain 5. What is the theoretical probability of rolling a 5 with 2 dice? Write your answer as a fraction and a percent. 4. Explain your answer to #3. 6. If you flip a coin 0 times, what is the theoretical probability of flipping heads? Write your answer as a fraction and a decimal. 7. Flip a coin 0 times. What is the experimental probability of flipping heads? Write your answer as a fraction and a decimal. 8. What is the difference between theoretical and experimental probability? 28 MHR Probability

27 It s On the Cup. A coffee shop promotion offers prizes in specially marked cups. The chance of winning is in 9. a) In your own words, explain the meaning of The chance of winning is in 9. b) List 2 words that mean the same as chance. c) What experiment that you completed recently has the same theoretical probability as getting a winning cup? 2. a) How can you simulate the coffee cup promotion without actually using coffee cups? To simulate means to model with an experiment. Describe or draw what you will do. Check out the table you completed on page 23. Go to pages 2 to write the definition for simulate in your own words. b) If you run this simulation 00 times, how many winners should you get? c) Explain how you determined your answer to part b) Free Coffee MHR 29

28 d) Test your hypothesis. Do the simulation exactly 00 times. Tally the results below. Winner Non-Winner e) Write your winning results 3 ways. As a percent of the total: As a fraction of the total: As a decimal: f) Some people like to show data like this on a graph. Use a circle graph to display your results. g) Did your experiment match the theoretical probability of the promotion? YES NO h) If not, explain why. 30 MHR Probability

29 Another way to run this kind of simulation is to use a device that generates random numbers. A random number generator is a tool that picks numbers so that each number has an equal probability of coming up on each try. A graphing calculator can be set up to work as a random number generator. It can then model the previous experiment. Go to pages 2 to write the definition for random number generator in your own words. Tech Tip: Using the Random Number Generator in a TI-83/84 Graphing Calculator. Press MATH. Scroll right so that PRB is highlighted. 2. Press 5 to select 5:randInt(. The command randint( tells the calculator to generate random integers. 3. Type,9). Make sure there are no spaces between the characters. This tells the calculator to select numbers between and Press ENTER. An integer from to 9 will be displayed. Continue pressing ENTER to generate more random integers Free Coffee MHR 3

30 3. a) Select a target number from to 9 for this experiment. Every time the random generator comes up with that number, you are a winner. b) Use a random number generator to select exactly 00 numbers ranging from to 9. Tally the results below. Winner Non-Winner c) How many times did the number you selected in part a) appear? d) State your winning percent. e) Explain your results in terms of simulating winning a prize from the coffee promotion. 4. a) Repeat the experiment another 00 times. Tally the results below. Winner Non-Winner b) Add these results to you totals from #3b). How many times did the number you selected in #3a) appear? c) State your winning percent. d) Is your result closer to the theoretical probability you calculated in #2b)? YES NO e) Explain your answer to part b). 32 MHR Probability

31 5. a) Collect the number of winning simulations in #4b) from everyone in your class. Number of Winning Simulations for Each Member of the Class Total Number of Winners in the Class = b) What is the total number of random numbers generated by the class? c) Calculate the class s winning percent. Check Your Understanding. According to the theoretical probability of the promotion, how many winning results should your class have had? 2. Explain why your individual results and the whole class s results may have differed. 3. Is the coffee shop s ad accurate? Explain Free Coffee MHR 33

32 .6 What Are the Odds? Focus: probability, media, number sense Warm Up. The probability of picking the 7 of clubs from a deck of cards is in. 3. What is the probability of flipping tails, tails, tails with 3 coins? 2. The probability of picking any red card from a deck of cards is in. 4. Reduce the following fractions to lowest terms. a) 5 _ 0 b) 70 _ 00 What Are Odds? You flip a coin. The probability of flipping heads is # of chances of winning _ # of possible flips 2. Another way of showing this is :2. heads tails Go to pages 2 to write the definition for odds in your own words. The odds of flipping heads are # of chances of winning # of chances of losing _. Another way of showing this is :. heads tails This can be confusing because the term odds is often used in the media as another word for probability or chance. 34 MHR Probability

33 An ad such as the following really means that the probability of winning is in 0 (or 0%). The odds of winning would be :9. chance of winning chance of not winning. a) Calculate the odds of drawing a red card from a deck of cards. How many red cards are in the deck? How many not red cards are in the deck? = Odds are shown as a ratio. The odds are :. b) What are the odds of drawing a spade from a deck of cards? The odds are :. c) What are the odds of drawing an ace from a deck of cards? d) What are the odds of drawing a jack, queen, or king from a deck of cards? e) What are the odds of rolling a 3 with one die? f) What are the odds of flipping heads, heads with 2 coins? g) What are the odds of flipping tails, tails, tails with 3 coins? What Are the Odds? MHR 35

35 Samples The school principal wants to do a survey about starting and finishing the school day 3 hours later than the current start and end times. 5. a) This would make your school day start at and finish at. b) Explain why the principal might not wish to survey the entire population of the school. The principal decides to survey a sample of the school population. A sample is part of a population. A good sample represents the entire population. 6. The principal is trying to decide which of the following samples would best represent the school s population. Consider your school s population. Read the description of each proposed sample. Decide which ones are potentially good samples. Which ones are potentially bad samples? Proposed Sample a) Survey all of the grade 9s. b) Survey all of the teachers. c) Survey 0 students from each grade and ask 0 teachers. d) Survey 0% of the population. e) Survey only those old enough to vote. f) Survey 0% of the population of each grade, the teachers, and the other staff. g) Survey the students in the cafeteria. Good Sample Bad Sample What Are the Odds? MHR 37

36 7. Choose proposed sample you classified as a Bad Sample. Explain your thinking. 8. a) Describe a good sample of your school s population. b) Discuss your sample idea with several other students. Listen to their coaching to make sure that your sample plan represents the school s population. Revise your sample if necessary. c) Using the sample, conduct a small survey to determine whether the odds are likely or unlikely that your school s population is in favour of starting and finishing the school day 3 hours later. Record your results. In Favour Not in Favour d) What can you conclude from your survey? Probability in the Media Many people read long-term forecasts before making plans. Can we play volleyball outside on Monday? Will it be warm enough to ride our bikes on Wednesday? Should we plan a weekend beach party? The long-term forecast on the next page shows the type of information the media provide. 38 MHR Probability

37 Long-Term Forecast Monday Sept. 3 Tuesday Sept. 4 Wednesday Sept. 5 Thursday Sept. 6 Friday Sept. 7 Saturday Sept. 8 P.O.P. High Low 24-Hr Rain Cloudy With Sunny Breaks Showers Isolated Showers Mostly Sunny Sunny Sunny 40% 80% 60% 20% 20% 0% 8 C 6 C 7 C 8 C 2 C 22 C C 3 C 9 C 4 C 6 C 8 C close to mm close to 0 mm close to 5 mm 9. a) What does P.O.P. stand for? b) How can it help you plan outdoor jobs or events? Hint: P.O.P. has two Ps. One stands for the topic of this chapter. The other is another word for rain. c) Which day, in your opinion, would be best for a family barbecue? Explain why. d) You work for a company that paves driveways. List the days you think you will be able to work this week. Check Your Understanding. Jack says, The odds of a 6-day forecast being right are slim to none. What might he mean by this? What Are the Odds? MHR 39

38 Review Date. Define theoretical probability. 2. What is the probability of each of the following? a) picking the 9 of clubs from a deck of cards (fraction) b) flipping heads with a coin (decimal) c) picking a diamond from a deck of cards (percent) d) rolling a 3 with die (fraction) e) rolling an even number with die (decimal) f) flipping heads or tails with a coin (percent) 3. a) How many combinations can be obtained by rolling 2 dice? b) List all of the combinations for rolling a 7 with 2 dice. c) Write the probability of rolling a 7 as a fraction of the total. 4. Define experimental probability. 5. Pick 0 cards from a deck of 52. a) How many spades did you pick? b) Write the number of spades you got as a fraction, a decimal, and a percent. = = fraction decimal percent 40 MHR Probability

39 6. Complete the table. Fraction Decimal Percent a) _ 2 b) _ 0 c) 0.3 d) 0.7 e) 90% f) 95% 7. a) Create and label a bar graph for the perfect world results for obtaining each suit when you cut a deck of cards 40 times. b) The graph in part a) shows probability. 8. A department store offers scratch and win tickets to its customers. The store claims that 25% of the tickets result in customers paying no taxes on purchases. a) Write the probability of getting a winning ticket as a fraction. b) If the store prints tickets, how many winning tickets are there? c) What are the odds of getting a winning ticket? Review MHR 4

40 Practice Test Date. Explain the difference between theoretical probability and experimental probability. 2. What is the theoretical probability of each of the following? a) picking a club from a deck of cards (fraction) b) picking a spade or a heart from a deck of cards (fraction) c) flipping tails with a coin (percent) d) rolling a 7 with die (percent) e) rolling an odd number with die (decimal) 3. a) How many combinations can you get by rolling 2 dice? b) List all of the combinations for rolling 0,, or 2 with 2 dice. c) Write the probability of rolling a 0 or greater as a fraction of the total. d) Write the answer to part c) in lowest terms. 4. Roll a die 20 times. a) How many 6s did you roll? b) Write the number of 6s that you rolled as a fraction, a decimal, and a percent. = = fraction decimal percent c) This is an example of probability. 42 MHR Probability

41 5. Complete the table. Fraction Decimal Percent a) _ 4 b) _ 5 c) 0.4 d) 0.65 e) 80% 6. Create and label a bar graph for the perfect world results for rolling 2 dice exactly 36 times. What totals do you get? 7. a) You flip 4 coins at the same time. What different ways can the coins land? List all combinations. b) What is the probability of getting all heads with 4 coins? Explain how you know Practice Test MHR 43

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