Probability Black Line Masters

Transcription

1 Probability Black Line Masters Draft (NSSAL) C. David Pilmer 2008

2 This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. The following are permitted to use and reproduce this resource for classroom purposes. Nova Scotia instructors delivering the Nova Scotia Adult Learning Program Canadian public school teachers delivering public school curriculum Canadian nonprofit tuition-free adult basic education programs The following are not permitted to use or reproduce this resource without the written authorization of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. Upgrading programs at post-secondary institutions Core programs at post-secondary institutions Public or private schools outside of Canada Basic adult education programs outside of Canada Individuals, not including teachers or instructors, are permitted to use this resource for their own learning. They are not permitted to make multiple copies of the resource for distribution. Nor are they permitted to use this resource under the direction of a teacher or instructor at a learning institution.

3 Introduction: This Nova Scotia School for Adult Learning resource is designed for Level II Mathematics. It is comprised of black line masters that instructors may wish to use with their learners when they are learning about probability. These are individual activity sheets. Collectively the sheets do not provide a comprehensive introduction to probability. Instructors should use other resources in conjunction with these black line masters. Table of Contents Theoretical Probabilities (Part I) Answers. 6 Theoretical Probabilities (Part II).. 7 Answers. 0 Experimental Probability (Part ).. Experimental Probability (Part 2). 3 Answers. 22 What are the Chances?. 24 Answers. 27 Comparing Theoretical and Experimental Probability.. 28 Answers. 33 Real World Probabilities Answers. 38 The Lotto 649 Activity 40 NSSAL i Draft

4 Theoretical Probability (Part ) Probability is the likelihood or chance of an event happening. When we look at the flipping of a coin, there are two outcomes: head, tail. The probability of obtaining a tail with a single flip of a coin can be found using the following formula. theoretica l probability = theoretical probability = 2, 0.50 or 50% number of favourable outcomes total number of possible outcomes The probability of obtaining a tail when flipping a coin once is 2, 0.50 or 50%. This is referred to as the theoretical probability. The theoretical probability is a measure of the likelihood of an event, based on calculations. Experimental probability is the measure of the likelihood of an event based on data from an experiment. If you flipped a coin 0 times, you would expect to obtain tails 5 times but this is not guaranteed. You might obtain tails 4 times. That would give you an experimental probability of 0.4. It is important to note that even though the theoretical probability of getting a tail was 0.5, the experimental probability turned out to be 0.4. This activity sheet is concerned with theoretical probability. Remember we calculate theoretical probability using the following formula. number of favourable outcomes theoretica l probability = total number of possible outcomes Example : What is the theoretical probability of obtaining a heart when you draw one card from a deck of cards? Answer: Number of favourable outcomes: 3 (3 hearts in a deck of cards) Total number of possible outcomes: 52 (52 cards in a deck) 3 Theoretical Probability =,, 0.25 or 25% 52 4 Example 2: What is the theoretical probability of obtaining a face card (King, Queen, Jack) when you draw one card from a deck of cards? Answer: Number of favourable outcomes: 2 (2 face cards in a deck of cards) Total number of possible outcomes: 52 (52 cards in a deck) 2 Theoretical Probability =, 0.23 or 23% 52 NSSAL Draft

5 Example 3: What is the theoretical probability of obtaining an odd number when you spin the following spinner once? 3 2 Answer: Number of favourable outcomes: 2 (2 odd numbers: and 3) Total number of possible outcomes: 3 (3 numbers) Theoretical Probability = 3 2, 0.67 or 67% Example 4: What is the theoretical probability of obtaining a number less than 3 when you roll a six-sided die once? Answer: Number of favourable outcomes: 2 (2 numbers less than 3: and 2) Total number of possible outcomes: 6 (6 numbers of a six-sided die) Theoretical Probability = 6 2, 3, 0.33 or 33% Example 5: What is the theoretical probability of obtaining a number greater than 6 when you roll a six-sided die once? Answer: Number of favourable outcomes: 0 (none; there are no numbers greater than 6) Total number of possible outcomes: 6 (6 numbers of a six-sided die) Theoretical Probability = 6 0, 0 or 0% Example 6: What is the theoretical probability of obtaining a number between, and including, and 8 when you roll an eight-sided die once? Answer: Number of favourable outcomes: 8 Total number of possible outcomes: 8 (8 numbers of a six-sided die) Theoretical Probability = 8 8, or 00% NSSAL 2 Draft

6 Questions: Express all probabilities as a fraction, decimal and percent.. Fill in the blanks with the appropriate numbers. (a) What is the theoretical probability of obtaining a 3, 4, or 5 with a single roll of a six-sided die? Number of favourable outcomes: Total number of possible outcomes: Theoretical Probability = (b) What is the theoretical probability of obtaining a 2 with a single roll of an eight-sided die? Number of favourable outcomes: Total number of possible outcomes: Theoretical Probability = (c) What is the theoretical probability of obtaining a number that is divisible by 3 with a single roll of a twenty-sided die? Number of favourable outcomes: Total number of possible outcomes: Theoretical Probability = (d) What is the theoretical probability of obtaining a king when dealt one card from a deck of cards? Number of favourable outcomes: Total number of possible outcomes: Theoretical Probability = (e) What is the theoretical probability of obtaining a king of hearts or diamonds when dealt one card from a deck of cards? Number of favourable outcomes: Total number of possible outcomes: Theoretical Probability = NSSAL 3 Draft

7 (f) What is the theoretical probability of obtaining an ace, 2, 3, or 4 when dealt one card from a deck of cards? Number of favourable outcomes: Total number of possible outcomes: Theoretical Probability = 2. Given the following spinner, find the probability of each event on a single spin (a) The number 5 is obtained. (b) An even number is obtained. (c) A number divisible by 3 is obtained. (d) Any number other than 2 is obtained. (e) A number greater than 9 is obtained. (f) The number 3 or 7 is obtained. (g) Any number between, and including, and 9 is obtained. 3. The theoretical probability of winning grand prize in Lotto 649 is (a) How many favourable outcomes are there to win the grand prize in Lotto 649? (b) What is the total number of possible outcomes in Lotto 649? (c) Knowing this information, would you take $20 and spend it on Lotto 649 tickets each week? Explain. NSSAL 4 Draft

8 4. (a) If we have a fair six-sided die, can we say that each possible outcome (, 2, 3, 4, 5, or 6) is equally likely to occur? (b) If we have a fair coin, can we say that each possible outcome (head or tail) is equally likely to occur? (c) If we had a spinner of the following configuration, can we say that each possible outcome (A, B, C, or D) is equally likely to occur? D C A B (d) If you look back at questions, 2 and 3, did we focus on questions that dealt with possible outcomes that are equally likely to occur or not? 5. What are your feelings about betting on games of chance (examples: poker, slot machines, roulette)? NSSAL 5 Draft

9 Answers: Please note that many of the decimal and percent answers had to be rounded off.. (F.O. - number of favourable outcomes, P.O. - total number of possible outcomes) 3 (a) F.O. = 3, P.O. = 6, Probability:,, 0.50 or 50% 6 2 (b) F.O. =, P.O. = 8, Probability: 8, 0.3 or 3% 6 3 (c) F.O. = 6, P.O. = 20, Probability:,, 0.30 or 30% 20 0 (d) F.O. = 4, P.O. = 52, Probability: 52 4, 3, 0.08 or 8% 2 (e) F.O. = 2, P.O. = 6, Probability:,, 0.04 or 4% (f) F.O. = 6, P.O. = 52, Probability:,, 0.3 or 3% (a) 9, 0. or % (b) 9 4, 0.44 or 44% (c) 9 3, 3, 0.33 or 33% (d) 9 8, 0.89 or 89% (e) 9 0, 0 or 0% (f) 9 2, 0.22 or 22% (g) 9 9, or 00% 3. (a) favourable outcome (b) possible outcomes (c) no 4. (a) yes (b) yes (c) no (d) equally likely NSSAL 6 Draft

10 Theoretical Probability (Part II) If you completed the activity sheet called Theoretical Probability (Part I), you learned to use the following formula in situations where each outcome was equally likely to occur. For example, with a six-sided die where each side is numbered through 6, each outcome (, 2, 3, 4, 5, or 6) is equally likely to occur. number of favourable outcomes theoretica l probability = total number of possible outcomes This changes if the sides of the die are numbered differently. Suppose three of the sides were numbered as and the remaining three sides were numbered 2 through 4. That means the chance of rolling a is three times greater than rolling a 2. The chance of rolling a is three times greater than rolling a 3. The chance of rolling a is three times greater than rolling a 4. Each outcome (, 2, 3, or 4) does not share the same likelihood of occurring Theoretical Probability of Rolling a One = 6 3, 2, 0.50 or 50% Theoretical Probability of Rolling a Two = 6, 0.7 or 7% Example : You have a bag that contains 2 red marbles, 3 green marbles and 5 yellow marbles. (a) If a single draw is made from the bag, what is the probability of obtaining a red marble? (b) If a single draw is made from the bag, what is the probability of obtaining a green marble? (c) If a single draw is made from the bag, what is the probability of obtaining a yellow marble? (d) Does each outcome (red marble, green marble, yellow marble) share the same likelihood of occurring? Explain. (e) If a single draw is made from the bag, what is the probability of obtaining a green or red marble? (f) If a single draw is made from the bag, what is the probability of obtaining a blue marble? Answers: (a) 0 2, 5, 0.20 or 20% (b) 0 3, 0.30 or 30% (c) 0 5, 2, 0.50 or 50% (d) Each outcome does not share the same likelihood of occurring because the probabilities differ. The probability of obtaining a yellow marble (0.50) is higher than the probability of obtaining a green marble (0.30). The probability of obtaining a green marble (0.30) is higher than the probability of obtaining a red marble (0.20). (e) 0 8, or 80% (f) 0 0, 0 or 0% NSSAL 7 Draft

11 Questions: Express all probabilities as a fraction, decimal and percent.. You have a six-sided die where two sides are numbered as, three sides are numbered as 2, and one side is numbered as (a) What is the probability of rolling a on a single roll? (b) What is the probability of rolling a 2 on a single roll? (c) What is the probability of rolling a 3 on a single roll? (d) What is the probability of rolling a 2 or 3 on a single roll? (e) Does each outcome (, 2, or 3) share the same likelihood of occurring? (f) What is the probability of rolling a, 2 or 3 on a single roll? (g) What is the probability of rolling a 4 or 5 on a single roll? 2. You have a six-sided die where each side is numbered through 6. The die is not a cube. It s a rectangular-based prism (see diagram). Do all six possible outcomes share the same likelihood of occurring? Explain. 3. You have a bag that contains 6 red marbles, 2 green marbles and 2 blue marbles. You will be making a single draw from the bag. (a) What is the probability of obtaining a red marble? (b) What is the probability of obtaining a green marble? (c) What is the probability of obtaining a blue marble? (d) Does each outcome (red marble, green marble, blue marble) share the same likelihood of occurring? (e) What is the probability of obtaining a purple marble? (f) What is the probability of obtaining a red or blue marble? (g) What is the probability of obtaining a red, green or blue marble? NSSAL 8 Draft

12 4. Is the following statement true or false? Explain. The probability of having a birthday in June is the same as the probability of having a birthday in any other month. 5. You have a spinner of the following configuration. B A C (a) What is the probability of obtaining A on a single spin? (b) What is the probability of obtaining B on a single spin? (c) What is the probability of obtaining A or C on a single spin? (d) What is the probability of obtaining D on a single spin? (e) What is the probability of obtaining A, B, or C on a single spin? 6. You are taking random shots (no aiming) at the following square dart board where each square has specific point values. These values are marked on the board. (a) What is the probability of obtaining a on a single throw? (b) What is the probability of obtaining a 2 on a single throw? (c) What is the probability of obtaining a 4 on a single throw? (d) What is the probability of obtaining a, 2, or 3 on a single throw? NSSAL 9 Draft

13 Answers: Please note that many of the decimal and percent answers had to be rounded off. 2. (a),, 0.33, or 33% (b),, 0.50, or 50% 6 2 (c), 0.7, or 7% (d),, 0.67, or 67% 6 3 (e) no 6 (f),, or 00% 6 0 (f), 0, or 0% 6 2. No - This die is far more likely to rest on one of the four rectangular faces, rather than the two square faces (a),, 0.30, or 30% (b),, 0.60, or 60% (c) 20 2, 0, 0.0, or 0% (d) no (e), 0, or 0% (f),, 0.40, or 40% (g),, or 00% No - Some months have 30 days, others have 3 days, and February as 28 days (29 on leap year). 5. (a) 2, 0.50, or 50% (b) 4, 0.25, or 25% (c) 4 3, 0.75, or 75% (d) 0, or 0% (e), or 00% 3 6. (a), 0.2, or 2% (b), 0.44, or 44% (c), 0.28, or 28% (d), 0.72, or 72% NSSAL 0 Draft

14 Experimental Probability (Part ) Probability is the likelihood or chance of an event happening. Experimental probabilities are based on experiments where data is collected. For example, if you wanted to know Steve Nash s probability of making a three-point shot while playing college basketball, you could examine his college record. He made 263 three-point shots out of 656 attempts while playing for Santa Clara University. This data can be used to calculate the experimental probability. Experimental Probability number of times an outcome occurs = number of times the experiment is conducted = or 0.40 Example : (a) Roll a six-sided die 2 times. Record the number of times you roll a 5. Based on this experiment, calculate the experimental probability of rolling a 5 on a single roll of this die. (b) Roll a six-sided die 24 times. Record the number of times you roll a 5. Based on this experiment, calculate the experimental probability of rolling a 5 on a single roll of this die. (c) Roll a six-sided die 36 times. Record the number of times you roll a 5. Based on this experiment, calculate the experimental probability of rolling a 5 on a single roll of this die. or 40% Answers: (Answers will vary from student to student.) (a) When the die was rolled 2 times, I obtained a 5 only one time. number of times an outcome occurs Experimental Probability = number of times the experiment is conducted = or 0.08 or 8% 2 (b) When the die was rolled 24 times, I obtained a 5 three times. 3 Experiment al Probability = or 0.3 or 3% 24 (c) When the die was rolled 36 times, I obtained a 5 seven times. 7 Experiment al Probability = or 0.9 or 9% 36 Note: If you rolled the die 000 times or more, you would find that the experimental probability of rolling a 5 on a single roll would approach the value 0.7 or 7%. Questions:. (a) Take a coin and toss it 0 times. Record the number of times it comes up heads. Calculate the experimental probability of obtaining a heads with this coin on a single flip. NSSAL Draft

15 (b) Take a coin and toss it 40 times. Record the number of times it comes up heads. Calculate the experimental probability of obtaining a heads with this coin on a single flip. (c) Google search Shodor Interactive Activities Coin Toss. This particular online program allows you to simulate multiple tosses of a coin. Enter 200 into the Number of Tosses box and click on Toss em! The program will list the heads (H) and tails (T) you obtained in your 200 tosses. If you select Table from the Display Results, the number of heads and the number of tails will be displayed. Use these results to calculate the experimental probability of obtaining a heads on a single flip of a coin. (d) Using the Shodor coin toss program, simulate 000 tosses of a coin. Use these results to calculate the experimental probability of obtaining a heads on a single flip of a coin. (e) The theoretical probability of obtaining a head on a single toss of a coin is 0.5 or 50%. Which of your experimental probabilities is closest to this theoretical probability? 2. Take a six-sided die and roll it 20 times. Record the number of times that you obtain a or 2. Using the data you just collected, calculate the experimental probability of obtaining a or 2 on a single roll of a six-sided die. 3. Using a small piece of corrugated cardboard (at least 8 cm by 8 cm), a paper clip, and a thumbtack, construct a spinner that is divided into four equal sections that are lettered A through D. The paper clip will be the spinning arm and the thumbtack will hold the paper clip so that it can spin around the center. Spin the spinner 20 times recording the number of times it lands in section A. Use this data to calculate the experimental probability of landing in section A with one spin of this spinner. A C B D NSSAL 2 Draft

16 Experimental Probability (Part 2) Probability is the likelihood or chance of an event happening. Experimental probabilities are based on experiments where data is collected. For example, if you wanted to know the probability that a particular professional baseball player is going hit the ball the next time at bat, then you would look at the player s batting history. Suppose the player was at bat 278 times and hit safely 02 times. This data can be used to calculate the experimental probability. Experimental Probability = = number of times an outcome occurs number of times the experiment is conducted or 0.37 We could interpret this to mean that this player has a 37% chance of a safe hit the next time at bat. It could also mean that he has a 63% chance of not getting a safe hit the next time at bat. or 37% Example : Marcy wants to invent a board game for her children. She wants to use a die in the game but decides to use a eight-sided die, rather than the traditional six-sided die. Her die is a hexagonally-based prism (see diagram) where the numbers through 6 are on the six rectangular faces, and the numbers 7 and 8 are on the two hexagonal faces. (a) Roll the die 80 times and complete a tally/frequency chart where you record the number of times you roll each of the 8 numbers. (b) What is the experimental probability that you will roll a 4 on a single roll? (c) What is the experimental probability that you will roll a 2 on a single roll? (d) What is the experimental probability that you will roll a 7 on a single roll? (e) Which numbers have a greater likelihood of being rolled? Answers: (a) (Sample Data) Tally Frequency Roll a 5 Roll a 2 8 Roll a 3 7 Roll a 4 9 Roll a 5 8 Roll a 6 4 Roll a 7 7 Roll an 8 22 NSSAL 3 Draft

17 (b) Experiment al Probability = number of times an outcome occurs times the experiment is conducted 9 = or 0. or 80 number of Exp. Probability of Rolling a 4 % (c) Experiment al Probability = number of times an outcome occurs times the experiment is conducted 8 = or 0.0 or 80 number of Exp. Probability of Rolling a 2 0% (d) Experiment al Probability = number of times an outcome occurs times the experiment is conducted 7 = or 0.2 or 80 number of Exp. Probability of Rolling a 7 2% (e) The numbers 7 and 8 have the greatest likelihood of being rolled. Example 2: Marcus, Jason, and Hamid play basketball in a gentleman s league. The three are trying to figure out who is best at foul shots. Their foul shot data for the last few years is listed below. Use your knowledge of probability to determine which one of the three players is the best at foul shots. Number of Attempted Foul Shots Number of Successful Foul Shots Marcus 32 9 Jason 27 4 Hamid 29 2 Answer: Marcus: number of times an outcome occurs Experiment al Probability = number of times the experiment is conducted 9 = 32 or 0.59 or 59% Jason: 4 Experimental Probability = 27 or 0.52 or 52% Hamid: 2 Experimental Probability = or 0.72 or 72% 29 Since Hamid has the highest probability (72%) of successfully completing foul shots, then he is the best at foul shots. NSSAL 4 Draft

18 Example 3: Jacob went to the Environment Canada website and was able to collect the following data regarding the total rainfall in Halifax during the month of August from 980 to Year Total Rainfall (mm) Year Total Rainfall (mm) Year Total Rainfall (mm) no data (a) Create a tally/frequency chart where the classes are 0-25, 25-50, 50-75, 75-00, 00-25, 25-50, 50-75, and For example, the total rainfall in August 980 (30.9 mm) would fit in the class because 30.9 mm is between 25 mm and 50 mm. (b) What is the probability that the rainfall in August will be between 25 mm and 50 mm? (c) What is the probability that the rainfall in August will be between 50 mm and 75 mm? (d) What is the probability that the rainfall in August will be between 75 mm and 00 mm? (e) What is the probability that the rainfall in August will be between 50 mm and 75 mm? (f) How likely is it that Halifax will have between 50 mm and 75 mm of rainfall in August? Answers: (a) Class Tally Frequency NSSAL 5 Draft

19 (b) (c) Experiment al Probability = number of times an outcome occurs times the experiment is conducted number of 4 = or 0.5 or 5% 27 There is a 5% chance that the total rainfall in Halifax during August will be between 25 mm and 50 mm. Experiment al Probability = number of times an outcome occurs times the experiment is conducted number of = or 0.4 or 4% 27 There is a 4% chance that the total rainfall in Halifax during August will be between 50 mm and 75 mm. 2 (d) Experimental Probability = or 0.07 or 7% 27 There is a 7% chance that the total rainfall in Halifax during August will be between 75 mm and 00 mm. (e) Experimental Probability = or 0.04 or 4% 27 There is a 4% chance that the total rainfall in Halifax during August will be between 50 mm and 75 mm. (f) It is very unlikely that there will be between 50 mm and 75 mm of rainfall in August because the probability is so low (only 4%). Questions: Express all probabilities as a fraction, decimal and percent.. Andrea played 378 holes of golf this summer. She made par on 23 of those holes. What is the probability that Andrea will make par on a hole? 2. Of US college students who studied outside of the country, students studied in the United Kingdom. What is the probability that a US student studying outside of the country will study in the United Kingdom? NSSAL 6 Draft

20 3. As of March 24, 2006, a total of 86 people worldwide had become ill due to the bird flu. Of those, 22 people had gotten the flu in Thailand and 93 people had gotten it in Vietnam. (a) Determine the probability that a person with the bird flu had gotten it in Thailand. (b) Determine the probability that a person with the bird flu had gotten it in Vietnam. 4. In 2004, males and females were born in the US. (a) How many births occurred in 2004? (b) What was the experimental probability that a US child born in 2004 was a male? (c) What was the experimental probability that a US child born in 2004 was a female? 5. A new drug is being tested that is supposed to lower blood pressure. This drug is given 200 people and the results are listed below. Number of People Whose Blood Pressure Lowered 42 Number of People Whose Blood Pressure Remained the Same 38 Number of People Whose Blood Pressure Got Higher 20 (a) What is the experimental probability that a person s blood pressure would lower on this new drug? (b) What is the experimental probability that a person s blood pressure would rise on this new drug? NSSAL 7 Draft

21 (c) If you had high blood pressure, would you consider using this new drug? Explain. 6. Gavin worked in a department store that was having its annual sale. Every customer who was buying an item received a Scratch and Save card at the register. The person would scratch the card and reveal the discount the person would get on their purchase. The cards offered discounts of 25%, 5%, and 0%. Gavin wanted to know the probability that someone would receive a discount of 25% at the register. To help answer this question he decided to keep track of all the discounts that he awarded at his register. (a) Here s the data Gavin collected. Complete the third column in the table. 25% Discounts 5% Discounts 0% Discounts Tally (b) How many discounts did Gavin give out in total? Frequency (c) What is the experimental probability that a customer would receive a 25% discount? (d) What is the experimental probability that a customer would receive a 5% discount? (e) What is the experimental probability that a customer would receive a 0% discount? (f) If you can only afford to buy a television with a 25% discount, should you participate in this Scratch and Save sale? Explain. NSSAL 8 Draft

22 7. You are going to drop a spoon 30 times from a height of metre. You are going to record the number of times the spoon lands the right-side up and the number of times it lands upside down. (a) Record your data in the following tally chart. Lands Right-side Up Lands Upside Down Tally Frequency (b) How many times did you conduct the experiment? (c) How many times did the spoon land right-side up? (d) How many times did the spoon land upside down? (e) What is the probability that the spoon will land right-side up? (f) What is the probability that the spoon will land upside down? 8. Tylena conducted a fitness survey. She wanted to know the ages of people who exercised 3 or more times a week. (a) She collected the following data. Complete the last column of the tally/frequency chart. Ages Tally Frequency (b) How many pieces of data does Tanya have? (c) What is the probability that people who exercise 3 or more times a week are between, and including, the ages 5 and 24? NSSAL 9 Draft

23 (d) What is the probability that people who exercise 3 or more times a week are between, and including, the ages 25 and 34? (e) What is the probability that people who exercise 3 or more times a week are between, and including, the ages 55 and 64? 9. Jerry went to the Environment Canada website and was able to collect the following data regarding the total snowfall in Annapolis Royal during the month of February from 987 to Year Total Snowfall (cm) Year Total Snowfall (cm) Year Total Snowfall (cm) no data (a) Complete the following tally/frequency chart. Please note that the class 0 to 0 includes 0 but does not include 0. Class Tally Frequency NSSAL 20 Draft

24 (b) How many pieces of data does Jerry have? (c) What is the probability that the snowfall in February will be between 50 cm and 60 cm? (d) What is the probability that the snowfall in February will be between 20 cm and 30 cm? (e) What is the probability that the snowfall in February will be between 0 cm and 0 cm? (f) How likely is it that Annapolis Royal will have between 0 and 0 cm of snow in February? NSSAL 2 Draft

25 Answers: Please note that many of the decimal and percent answers had to be rounded off = = 0.33 = 33% = 0.25 = 25% (a) = = 0.2 = 2% (b) = = 0.50 = 50% (a) (b) = 0.5 = 5% (c) = 0.49 = 49% (a) = = 0.7 = 7% (b) 200 = 0 = 0.0 = 0% (c) You should consider the new drug because it is likely that it could lower your blood pressure. You know this because the probability of lowering the blood pressure is fairly high (7%). You should still make sure that you talk to your doctor about the new drug. 6. (a) Frequency 25% Discounts 4 5% Discounts 0% Discounts 27 (b) 42 (c) = = 0.0 = 0% 2 (d) = 0.26 = 26% (e) = 0.64 = 64% 42 (f) No, there is only a very slight chance (0%) that you would be able to get the discount you needed. 7. Answers will vary. NSSAL 22 Draft

26 8. (a) Ages Frequency (b) 5 8 (c) 5 = 0.2 = 2% 42 (d) = 0.28 = 28% 5 22 (e) = 0.5 = 5% 5 9. (a) Class Frequency (b) 9 (c) 9 5 = 0.26 = 26% (d) 9 6 = 0.32 = 32% (e) 9 = 0.05 = 5% (f) very unlikely NSSAL 23 Draft

27 What are the Chances? When someone gives you a probability, they are providing you with a number that describes the chance of a particular event occurring. A probability can be written as a fraction, decimal, or percent. If there is almost no chance that the event will occur, then the probability is very close to 0. For example, the probability of selecting a king of spades from a deck of cards is or approximately If there is no chance of the event occurring, then the 52 probability is equal to 0. For example, the probability that your instructor is an alien from a distant planet is 0. If there is an excellent chance that an event will occur, then the probability is very close to. For example, the probability of rolling two six-sided dice and getting a sum of 3 or 35 more on those dice is or approximately If you are certain that a particular event 36 will occur, then the probability is equal to. For example, the probability that there will be at least one birth in Nova Scotia in January is. The following scale provides the descriptive terms (above the line) and their corresponding probabilities (below the line). Impossible Unlikely Half of the Time Likely Certain 0 0% % % % % You will use this scale to answer the following questions. Questions:. Decide the chance of each one of these events occurring using the terms impossible, almost impossible, unlikely, half of the time, likely, almost certain, certain. (a) Event You flip a coin and get a head. Description (b) In an adult education classroom with twenty students at least one student is male. NSSAL 24 Draft

28 Event (c) You have a bag with 7 red marbles and 2 blue marbles. Without looking, you pull a red marble from the bag. (d) There will be a major snowstorm in Truro on August 2 of this year. (e) When you are dealt five cards from a deck of cards, you end up with four of a kind. (f) You roll a six-sided die and get a 5. Description (g) (h) (i) (j) (k) You have a safe plane trip when you travel from Halifax to Gander. You buy a Lotto 649 ticket and you do not win the grand prize. You buy a Lotto 649 ticket and you win the grand prize. A solid iron bar will sink when it is thrown into the lake. The next child born at the IWK will be a girl. (l) You roll a six-sided die and get a 3, 4, 5, or 6. (m) You roll an eight-sided die and get a or 2. (n) (o) (p) (q) (r) Having snowfall in the first week in January in Sydney. Having a month with 32 days. Having a birthday in the summer. Having a birthday in the spring or summer. Having a birthday in any other season than winter. 2. (a) Describe a situation that is almost certain. (b) Describe a situation that is unlikely. (c) Describe a situation that is impossible. NSSAL 25 Draft

29 3. Draw lines to connect the event to the appropriate probability. Event (a) It is almost impossible to get a royal flush (A, K, Q, J, 0 from the same suit) when dealt 5 cards from a deck. (b) Almost half of the time you should be able to pick a red ball from a bag containing 6 red balls and 7 green balls. (c) (d) (e) (f) It is impossible for one person to have two birthdays in the same year. It is unlikely that you would be able to pick a red ball from a bag containing 2 red balls and 3 green balls. It is likely that you will roll a number greater than 2 on an eight-sided die. It is certain that the sun will rise in the morning and set at night. Probability (i) 0 (ii) 6 8 (iii) (iv) (v) (vi) You have been supplied with a deck of cards, but all the face cards (Kings, Queens, Jacks) have been removed. Describe a situation, with these cards, that matches each description. (a) Describe a situation that is certain. (b) Describe a situation that is likely. (c) Describe a situation that occurs half of the time. (d) Describe a situation that is unlikely. (e) Describe a situation that is impossible. NSSAL 26 Draft

30 Answers:. (a) half of the time (b) almost certain (c) likely (d) impossible (e) almost impossible (f) unlikely (g) almost certain (h) almost certain (i) almost impossible (j) certain (k) half of the time (l) likely (m) unlikely (n) almost certain or likely (o) impossible (p) unlikely (q) half of the time (r) likely 2. Answers will vary. 3. (a) matches with (iv) (b) matches with (v) (c) matches with (i) (d) matches with (vi) (e) matches with (ii) (f) matches with (iii) 4. Answers will vary. Sample answers have been provided. (a) certain that a card drawn from the deck is not a face card (b) likely that a card drawn from the deck is 3 or greater (c) half of the time a diamond or heart will be drawn from the deck (d) unlikely that a card from the deck is a 9 or 0 (e) impossible for a face card to be drawn from the deck NSSAL 27 Draft

31 Comparing Theoretical and Experimental Probability As you have learned, theoretical probability is based on logic and is calculated using the following formula. number of favourable outcomes Theoretica l Probability = total number of possible outcomes For example, the theoretical probability of rolling a 2 on a six-sided die is 6 or 0.7 or 7%. Experimental probability is based on experiments where data is collected. This data and the following formula are used to determine the experimental probability. number of times an outcome occurs Experiment al Probability = number of times the experiment is conducted For example, suppose you rolled a six-sided die 30 times and it came up as a two 6 times. The experimental probability of rolling a 2 would be 30 6 or 5 or 0.20 or 20%. Notice that the theoretical probability of rolling a two and the experimental probability 6 5 of rolling a two were not the same. In the questions that follow, you will be comparing theoretical probability (what you expect to happen) and experimental probability (what actually happens). Questions: Express all probabilities as a fraction, decimal, and percent.. (a) What is the theoretical probability of getting a head on a single flip of a fair coin? (b) Monica flipped a coin 20 times and it came up heads 2 times. What is the experimental probability of getting a head on a single flip of this coin? (c) Take a coin and flip it 0 times. Record the number of times you get heads. Based on this data figure out the experimental probability of getting a head on a single flip of this coin? NSSAL 28 Draft

32 2. (a) What is the theoretical probability of getting a 5 on a single roll of a six-sided die? (b) What is the theoretical probability of getting a 3 on a single roll of a six-sided die? (c) Sonya rolled a six-sided die 60 times and obtained the following data. Frequency Roll a Roll a 2 9 Roll a 3 3 Roll a 4 9 Roll a 5 8 Roll a 6 0 What is the experimental probability of getting a 5 on a single roll of this die? (d) Using Sonya s data, figure out the experimental probability of getting a 3 on a single roll of this die? (e) Take a six-sided die and roll it 8 times. Record the number of times you roll a 5. Based on this data figure out the experimental probability of getting a 5 on a single roll of this die? 3. Six wooden blocks of the same size and shape are placed in a bag. One block is yellow. Two blocks are red. Three blocks are blue. (a) What is the theoretical probability of drawing a yellow on a single draw? (b) What is the theoretical probability of drawing a red on a single draw? (c) What is the theoretical probability of drawing a blue on a single draw? NSSAL 29 Draft

33 4. In the last question there was a bag with one yellow block, two red blocks, and three blue blocks. Andrea decided to conduct an experiment using this bag of blocks. She picked a block from the bag without looking, recorded the color, and then placed the block back in the bag. She did this 60 times. The table below shows the data she collected. Color of Block Frequency Yellow 9 Red 7 Blue 34 (a) What is the experimental probability of drawing a yellow on a single draw? (b) What is the experimental probability of drawing a red on a single draw? (c) What is the experimental probability of drawing a blue on a single draw? 5. Look at the theoretical and experimental probabilities that you got in the last two questions. Were the expected values (theoretical probabilities) and actual values (experimental probabilities) equal? Why is this so? 6. Google Search NCTM illuminations adjustable spinner. With this online simulation you can adjust the size and number of sections on the spinner. You can repeatedly use the spinner and examine the relationship between the theoretical probability and experimental probability. Set the Number of sectors to 3 using the - and + buttons. Set the Number of spins to 50. Set Blue at 50%, Cyan at 25%, and Green at 25% (then press Update). NSSAL 30 Draft

34 (a) Press Spin. The data for those 50 spins will appear in the table. Record this information. Count Experimental Theoretical Blue 50% Cyan 25% Green 25% (b) Press Spin a few more times until the Number of spins so far reads 250. The data for 250 spins now appears in the table. Record this information. Count Experimental Theoretical Blue 50% Cyan 25% Green 25% (c) Press Spin several more times until the Number of spins so far reads 000. The data for 000 spins now appears in the table. Record this information. Count Experimental Theoretical Blue 50% Cyan 25% Green 25% (d) For which experiment (50 spins, 250 spins, or 000 spins) did you get the best match between the experimental and theoretical probabilities? (e) If you were to use the spinner 8000 times, would you expect the theoretical probabilities and experimental probabilities to be closer or further apart? Check your answer using the online spinner. Press New experiment, set Number of spins to 8000, and press Spin. Did you get what you expected? (f) The law of large numbers states that probability statements apply in practice to a large number of trials. In the case of this spinner simulation, one spin is considered one trial. Look at your answers to questions (d) and (e). Do these answers support the law of large numbers? Explain. NSSAL 3 Draft

35 7. A fair coin was repeated tossed and the following data was collected. Decimal and percent probabilities have been rounded off in many cases. Number of Tosses Expected Number of Heads Actual Number of Heads Observed Experimental Probabilities or 0.40 or 40% or 0.59 or 59% or 0.48 or 48% or 0.5 or 5% Explain how the data above relates to the law of large numbers. 8. Google search maths online probability. Select What s in Santa s sack. In this game, you will draw items from and then return them to Santa s sack. You will decide how many times this will occur. You will use the results from this simulation to predict the number of toys of each type found in the sack. (a) When you get the correct answer, print the page. (b) How did the law of large numbers help you with this game? (c) What should the theoretical probability for the drawing of each toy. (d) We re the experimental probabilities the same. Why or why not? NSSAL 32 Draft

36 Answers: Many of the probabilities that were expressed as decimals and percents have been rounded off.. (a) or 0.50 or 50% (b) or or 0.60 or 60% 20 5 (c) Answers will vary. 2. (a) or 0.7 or 7% 6 (b) or 0.7 or 7% (c) or or 0.3 or 3% (d) or 0.22 or 22% 60 (e) Answers will vary. 3. (a) or 0.7 or 7% 6 2 (b) or or 0.33 or 33% (c) or or 0.50 or 50% (a) or or 0.5 or 5% (b) or 0.28 or 28% (c) or or 0.57 or 57% The theoretical and experimental probabilities are not equal to each other but the values are close to each other. Theoretical probability describes the likelihood something will occur. It is not a guarantee. That is why the theoretical probability (what you expect to happen) is close by not necessarily equal to the experimental probability (what actually happens). 6. (a), (b), and (c) - Answers will vary. (d) Typically the 000 spins will give the best match. (e) The experimental should be very close to the theoretical probability when the spinner is used 8000 times. NSSAL 33 Draft

37 (f) Notice that as we go from 50 spins to 250 spins, to 000 spins, and finally to 8000 spins, the experimental probabilities get closer to the theoretical probabilities. A larger number of trials (spins) produce a better match between the experimental and theoretical probabilities. This is exactly what the law of large numbers states. 7. As the number of tosses (trials) increases, the experimental probability gets closer to the theoretical probability of 0.50 or 50%. One can say that the theoretical statement of 0.50 or 50% applies to a large number of trials (tosses). This is the law of large numbers. 8. (b) If you don t have enough trials then your experimental probabilities will not be very close to the theoretical probabilities and it is unlikely that you will be able to figure out how many of each toy are hidden in the sack. You need a large number of trials. That means that you are using the law of large numbers to solve this problem. (c) Answers will vary based on which game you selected. (d) Probably not NSSAL 34 Draft

38 Real World Probability Over the last few lessons you ve looked at all sorts of theoretical and experimental probabilities. Many of the questions dealt with flipping coins, drawing cards, or spinning spinners. Although these types of questions are useful when trying to understand probability, these contexts don t represent.. Anne and Dave had one healthy child. They were thinking of having a second child when Anne learned that she had a connective tissue disorder. Her doctor told her that, due to this disorder, there was a 25% probability that her child would be born with significant heart defects. How concerned should Anne and Dave be about this new information? In your opinion, should they try to have a second child? Explain your reasoning. 2. Colin spent much of his youth doing outdoor sports where he had excessive exposure to the sun. At the age of 35, his doctor informed him that a small growth on his forehead was a form of skin cancer. The doctor told him that there was a 99% survival rate for this form of cancer. How concerned should Colin be about his chances for survival if he gets treatment for the cancer? Explain your reasoning. 3. Statistics Canada reported that in 2007 there were 7 homicides and 3866 assaults for every million people in Canada. (a) What was the probability of being the victim of a homicide in 2007? (b) What was the probability of a being the victim of an assault in 2007? (c) Are you concerned with the probabilities you calculated in questions (a) and (b). Would you change your way of living based on these probabilities? (d) Do you think that these probabilities can be applied to anyone living anywhere in Canada? Explain. NSSAL 35 Draft

39 4. The probability of sustaining a head injury in a motorcycle accident when a helmet is not worn is The probability of sustaining a head injury with a helmet is Do you agree with the law that all riders must wear helmets? Explain. 5. Go to YouTube. Search for the video titled probability %. View the video. What is the significance of using the number % in the title of the video? 6. In James Bond movies, James regularly performs incredible tasks. In many of his movies, he plays poker against the villain and ultimately wins. In one movie, his winning hand is a royal flush (Ace, King Queen, Jack and 0 of the same suit). The probability of being dealt these 5 cards is in If you were James, would you rest your own fate and that of your country on this kind of probability? Explain. 7. Did you know that in a group of 23 people, the probability of at least two people sharing the same birthday is slightly greater that 0.5. Are you surprised by this? What would happen if there were 00 people in the group? Explain your reasoning. 8. The chances of selecting the winning numbers with the purchase of one ticket in Lotto 649 is in Suppose you could buy tickets, all with different numbers? You would be guaranteed to win but would this be a wise decision? Why or why not? NSSAL 36 Draft

40 9. Do your best to match each of these situations to the appropriate probability. Some of these might surprise you. (a) Killed in Commercial Plane Crash in 500 (b) Die from AIDS in 6000 (c) Killed by a Hurricane in 000 (d) Die from Cancer in.4 million (e) Die in a Car Accident in 6 million (f) Killed by Lightning in 0 million 0. A major coffee shop chain has a promotion where there is a in 0 chance of winning a free muffin with the purchase of a coffee. If the company sells cups of coffee during the promotion, how many free muffins should they expect to give away?. Create your own real-world situation that might result in a probability of Angela knows that the probability of winning the grand prize in Lotto 649 is in She wants to try to try to visualize this. She is going to use Post-It notes to do this. She knows that a small stack of Post-It notes has 00 sheets and measures. cm in thickness. (a) She wants a stack of Post-It notes that has sheets in it. How thick (tall) will this stack be? Select the correct answer. (i).54 km tall (ii) 0.53 km tall (iii) 03.7 m tall (iv) 8.92 m tall (b) Her friend will place an X on one of the sheets to represent the grand prize. What is the likelihood that Angela will be able to randomly select that sheet with the X on it from stack of sheets? NSSAL 37 Draft

41 Answers:. There is a in 4 chance that the child will be born with a heart defect. That means that there is a 3 in 4 chance that the child will not have a heart defect. Some people would argue that since Anne and Dave already have one healthy child, that the risk is too high in having a second child. Others would state that the risk is low enough to justify trying to have a second child. There isn t a clear answer to this question. 2. A 99% survival rate is very good. It means that there is only a % chance that Colin will not survive the treatment for skin cancer. There is only a very slight possibility that he wouldn t survive the treatment. He shouldn t be very concerned but he should still seek immediate treatment (a) Probability = (b) Probability = (c) No, these probabilities are very low. (d) No, people living in more disadvantaged areas or engaging in illegal activities might be subject to higher probabilities of assault and/or homicide. 4. The probability of sustaining a head injury when no helmet is worn is 49% or almost in 2. That is an alarming probability. That probability reduced significantly when a helmet is worn. When you consider the burden to the family of the accident victim, the victim s quality of life, and the cost to health care system, it seems that logical that helmets should be worn. Although helmets do not ensure complete safety, they do reduce the risk significantly. 5. It is virtually impossible for this sequence of events to occur that is why it is titled probability %. It seems to be an appropriate title. You can probably say that is sequence of events would never occur. If so, maybe it should be titled probability 0%. 6. The probability of in is very close to 0%. That means that it is almost impossible to be dealt a royal flush. It hardly seems worsh it for James to risk his own safety and the security of his country on such a slim chance. 7. As the number of people increases, the probability of having two people with the same birthday increases. 8. You would win but there is still a possibility that you would have to share your winning with someone else who selected the same numbers. You also have the problem that you would initially need $ to buy the tickets. If you had over $3 million dollars, would you want to risk it in this manner? There is also the problem of buying over 3 million tickets. Even if it took only 5 seconds to purchase a single ticket, it would take over 809 days (2.2 years) to purchase over 3 million tickets. Lotto 649 draws occur every week. You don t have enough time to buy the 3 million tickets. NSSAL 38 Draft

42 9. (d) in 500 (e) in 6000 (b) in 000 (f) in.4 million (c) in 6 million (a) in 0 million muffins 2. (a) (i).54 km tall (b) almost impossible NSSAL 39 Draft

43 The Lotto 649 Activity Have you ever played Lotto 649? Yes No If you answered yes, do you play it regularly? Yes No If you play it regularly, how much do you spend per month on Lotto 649? Less than $5 Between $5 and $5 More than $5 You are now going to select 6 numbers between and 49. Do not select the same number twice. Put these 6 numbers from small to largest in the first six boxes below. From the remaining 43 numbers that you did not choose, choose one more. This will be your bonus number. Fill it in the last box. Bonus Number These numbers will be your Lotto 649 numbers. Your instructor will now supply you with sheets that have all the winning numbers for Lotto 649 from June 982 until June 2008 (Obtained from ). That is 26 years of Lotto 649 winning numbers from 2528 drawings. Your mission is to compare your numbers to the winning numbers on the sheets and see if you would have won anything. Record the information in the table below. Matches Number of Times the Matches Occurred Dates that the Matches Occurred Prize (Fictitious) All 6 numbers match $ out of 6 plus the bonus number $ out of 6 matches $ out of 6 matches $75 3 out of 6 matches $0 2 out of 6 plus the bonus number $5 NSSAL 40 Draft

44 You bought 2528 tickets over the 26 years. If each ticket costs $2, how much would you have spent in total? How much money did you win? For Your Information Here are the probabilities of winning each of the prizes in Lotto 649. Probability of Matching 6 of 6 (Grand Prize) = Probability of Matching 5 of 6 plus Bonus # = or Probability of Matching 5 of 6 = or approximately Probability of Matching 4 of 6 = or approximately Probability of Matching 3 of 6 = or approximately Probability of Matching 2 of 6 plus Bonus # = or approximately Total Number of Potential Prizes = = Total Number of Different Tickets = Probability of Winning a Prize = or approximately 0.03 or 3% That means that you lose approximately 97% of the time when you play Lotto 649. Watch the following YouTube video. &feature=plpp_video (or Google Search: YouTube Understanding the Chances of Winning Lotto 6 49 Pilmer) NSSAL 4 Draft