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1 PAGES 45 KEY Organize the data into the circles. A. Factors of 64: 1, 2, 4, 8, 16, 32, 64 B. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 A B Answer Questions about the diagram below Fall Sports Winter Sports Spring Sports 1) How many students play sports yearround? 3 2) How many students play sports only in the spring and fall? 6 3) How many students play sports only in the winter and fall? 13 4) How many students play sports only in the winter and spring? 2 5) How many students play only one sport? 48 6) How many students play at least two sports? 24
2 7) Suppose you have a standard deck of 52 cards. Let: a. Describe for this experiment, and find the probability of. A B = {7 spades, 7 clubs, 7 hearts, all diamonds} P(A B) = 16/52 or 4/13 b. Describe for this experiment, and find the probability of. A B = {7 diamonds} P(A B) = 1/52 8) Suppose a box contains three balls, one red, one blue, and one white. One ball is selected, its color is observed, and then the ball is placed back in the box. The balls are scrambled, and again, a ball is selected and its color is observed. What is the sample space of the experiment? S = {RR, RB, RW, BR, BB, BW, WR, WB, WW} 9) Suppose you have a jar of candies: 4 red, 5 purple and 7 green. Find the following probabilities of the following events: Selecting a red candy. 1/4 Selecting a purple candy. 5/16 Selecting a green or red candy. 11/16 Selecting a yellow candy. 0 Selecting any color except a green candy. 9/16 Find the odds of selecting a red candy. 7/9 Find the odds of selecting a purple or green candy. 9/7 10) What is the sample space for a single spin of a spinner with red, blue, yellow and green sections spinner? {red, blue, yellow, green} What is the sample space for 2 spins of the first spinner? {RR, RB, RY, RG, BR, BB, BY, BG, YR, YB, YY, YG, GR, GB, GY, GG} If the spinner is equally likely to land on each color, what is the probability of landing on red in one spin? 1/4 What is the probability of landing on a primary color in one spin? 3/4 What is the probability of landing on green both times in two spins? 1/16 11) Consider the throw of a die experiment. Assume we define the following events: Describe for this experiment. {1,2,3,4,6} Describe for this experiment. {2} Calculate and, assuming the die is fair. 5/6 1/6
3 PAGES KEY Independent and Dependent Events 1. Determine which of the following are examples of independent or dependent events. a. Rolling a 5 on one die and rolling a 5 on a second die. independent b. Choosing a cookie from the cookie jar and choosing a jack from a deck of cards. Indepen. c. Selecting a book from the library and selecting a book that is a mystery novel. Dependent d. Going to the beach and bringing an umbrella. Dependent e. Getting gasoline for your car and getting diesel fuel for your car. dependent f. Choosing an 8 from a deck of cards, replacing it, and choosing a face card. Indepen. g. Choosing a jack from a deck of cards and choosing another jack, without replacement. dependent h. Being lunchtime and eating a sandwich. dependent 2. A coin and a die are tossed. Calculate the probability of getting tails and a 5. 1/12 3. In Tania's homeroom class, 9% of the students were born in March and 40% of the students have a blood type of O+. What is the probability of a student chosen at random from Tania's homeroom class being born in March and having a blood type of O+?.036 or 3.6% 4. If a baseball player gets a hit in 31% of his atbats, what it the probability that the baseball player will get a hit in 5 atbats in a row?.0029 or.29% 5. What is the probability of tossing 2 coins one after the other and getting 1 head and 1 tail? 1/ cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is the probability that they both will be clubs? 1/ cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is the probability that they both will be face cards? 9/ If the probability of receiving at least 1 piece of mail on any particular day is 22%, what is the probability of not receiving any mail for 3 days in a row? P(not receiving mail) = =.78. P(no mail for 3 days) = (.78)(.78)(.78) =.475 or 47.5% 9. Johnathan is rolling 2 dice and needs to roll an 11 to win the game he is playing. What is the probability that Johnathan wins the game? 2/6 x 1/6 = 1/ Thomas bought a bag of jelly beans that contained 10 red jelly beans, 15 blue jelly beans, and 12 green jelly beans. What is the probability of Thomas reaching into the bag and pulling out a blue or green jelly bean and then reaching in again and pulling out a red jelly bean? Assume that the first jelly bean is not replaced. 27/37 x 10/36 = 15/74
4 11. For question 10, what if the order was reversed? In other words, what is the probability of Thomas reaching into the bag and pulling out a red jelly bean and then reaching in again and pulling out a blue or green jelly bean without replacement? 10/37 x 27/36 = 15/74; same 12. What is the probability of drawing 2 face cards one after the other from a standard deck of cards without replacement? 12/52 x 11/51 = 11/ There are 3 quarters, 7 dimes, 13 nickels, and 27 pennies in Jonah's piggy bank. If Jonah chooses 2 of the coins at random one after the other, what is the probability that the first coin chosen is a nickel and the second coin chosen is a quarter? Assume that the first coin is not replaced. 13/50 x 3/49 = 39/2450 or For question 13, what is the probability that neither of the 2 coins that Jonah chooses are dimes? Assume that the first coin is not replaced. 43/50 x 42/49 = 129/ Jenny bought a halfdozen doughnuts, and she plans to randomly select 1 doughnut each morning and eat it for breakfast until all the doughnuts are gone. If there are 3 glazed, 1 jelly, and 2 plain doughnuts, what is the probability that the last doughnut Jenny eats is a jelly doughnut? 5/6 x 4/5 x 3/4 x 2/3 x 1/2 x 1 = 1/6 16. Steve will draw 2 cards one after the other from a standard deck of cards without replacement. What is the probability that his 2 cards will consist of a heart and a diamond? 13/52 x 13/51 = 13/204
5 PAGES KEY Mutually Exclusive and Inclusive Events 1. 2 dice are tossed. What is the probability of obtaining a sum equal to 6? 5/ dice are tossed. What is the probability of obtaining a sum less than 6? 10/36 or 5/ dice are tossed. What is the probability of obtaining a sum of at least 6? 26/36 or 13/18 4. Thomas bought a bag of jelly beans that contained 10 red jelly beans, 15 blue jelly beans, and 12 green jelly beans. What is the probability of Thomas reaching into the bag and pulling out a blue or green jelly bean? 27/37 5. A card is chosen at random from a standard deck of cards. What is the probability that the card chosen is a heart or spade? Are these events mutually exclusive? ½; yes 6. 3 coins are tossed simultaneously. What is the probability of getting 3 heads or 3 tails? Are these events mutually exclusive? 2/8 or ¼; yes 7. In question 6, what is the probability of getting 3 heads and 3 tails when tossing the 3 coins simultaneously? 0 8. Are randomly choosing a person who is lefthanded and randomly choosing a person who is righthanded mutually exclusive events? Explain your answer. Answers will vary; Most will say yes because people are either left handed or right handed, but some may say no because some people are ambidextrous. 9. Suppose 2 events are mutually exclusive events. If one of the events is randomly choosing a boy from the freshman class of a high school, what could the other event be? Explain your answer. Randomly choosing a girl from the freshman class 10. Consider a sample set as. Event is the multiples of 4, while event is the multiples of 5. What is the probability that a number chosen at random will be from both and? P(A and B) = P(A) x P(B) = 5/10 x 2/10 = 1/ For question 10, what is the probability that a number chosen at random will be from either or? P(A or B) = P(A) + P(B) P(A and B) = 5/10 + 2/10 1/10 = 6/10 or 3/5 12. Jack is a student in Bluenose High School. He noticed that a lot of the students in his math class were also in his chemistry class. In fact, of the 60 students in his grade, 28 students were in his math class, 32 students were in his chemistry class, and 15 students were in both his math class and his chemistry class. He decided to calculate what the probability was of selecting a student at random who was either in his math class or his chemistry class, but not both. Draw a Venn diagram and help Jack with his calculation. Class Math Chemistry Brenda did a survey of the students in her classes about whether they liked to get a candy bar or a new math pencil as their reward for positive behavior. She asked all 71 students she taught, and 32 said they would like a candy bar, 25 said they wanted a new pencil, and 4 said they wanted both. If Brenda were to select a student at random from her classes, what is the probability that the student chosen would want: a. a candy bar or a pencil? 32/ /71 4/71 = 53/71 b. neither a candy bar nor a pencil? 18/ A card is chosen at random from a standard deck of cards. What is the probability that the card chosen is a heart or a face card? Are these events mutually inclusive? 13/ /52 3/52 = 22/52 or 11/26; yes
6 15. What is the probability of choosing a number from 1 to 10 that is greater than 5 or even? 5/10 + 5/10 3/10 = 7/ A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the letters in the word ENGLISH on it or randomly choosing a tile with a vowel on it? 7/26 + 5/26 2/26 = 10/26 or 5/ Are randomly choosing a teacher and randomly choosing a father mutually inclusive events? Explain your answer. Yes, some teachers are also fathers. 18. Suppose 2 events are mutually inclusive events. If one of the events is passing a test, what could the other event be? Explain your answer. Answers will vary. One answer could be getting an A on the test.
7 PAGES KEY Conditional Probability 1. Compete the following table using sums from rolling two dice. Us e the table to answer questions fair dice are rolled. What is the probability that the sum is even given that the first die that is rolled is a 2? 1/ fair dice are rolled. What is the probability that the sum is even given that the first die rolled is a 5? 1/ fair dice are rolled. What is the probability that the sum is odd given that the first die rolled is a 5? 1/2 5. Steve and Scott are playing a game of cards with a standard deck of playing cards. Steve deals Scott a black king. What is the probability that Scott s second card will be a red card? 26/51 6. Sandra and Karen are playing a game of cards with a standard deck of playing cards. Sandra deals Karen a red seven. What is the probability that Karen s second card will be a black card? 26/51 7. Donna discusses with her parents the idea that she should get an allowance. She says that in her class, 55% of her classmates receive an allowance for doing chores, and 25% get an allowance for doing chores and are good to their parents. Her mom asks Donna what the probability is that a classmate will be good to his or her parents given that he or she receives an allowance for doing chores. What should Donna's answer be?.25/.55 = 45.5% 8. At a local high school, the probability that a student speaks English and French is 15%. The probability that a student speaks French is 45%. What is the probability that a student speaks English, given that the student speaks French?.15/.45 = 33.3% 9. On a game show, there are 16 questions: 8 easy, 5 mediumhard, and 3 hard. If contestants are given questions randomly, what is the probability that the first two contestants will get easy questions? P(2 nd is easy 1 st is easy) = 1/2 10. On the game show above, what is the probability that the first contestant will get an easy question and the second contestant will get a hard question? P(2 nd is hard 1 st is easy) = 1/5 11. Figure 2.2 shows the counts of earned degrees for several colleges on the East Coast. The level of degree and the gender of the degree recipient were tracked. Row & Column totals are included.
8 a. What is the probability that a randomly selected degree recipient is a female? 714/1375 = 51.9% b. What is the probability that a randomly chosen degree recipient is a man? 661/1375 = 48.1% c. What is the probability that a randomly selected degree recipient is a woman, given that they received a Master's Degree? 128/293 = 43.7% d. For a randomly selected degree recipient, what is P(Bachelor's Degree Male)? 438/661 = 66.3% 12. Animals on the endangered species list are given in the table below by type of animal and whether it is domestic or foreign to the United States. Complete the table and answer the following questions. Mammals Birds Reptiles Amphibians Total United States Foreign Total An endangered animal is selected at random. What is the probability that it is: a. a bird found in the United States? 78/663 = 11.8% b. foreign or a mammal? 498/ / /663 = 84.6% c. a bird given that it is found in the United States? 14/165 = 8.5% d. a bird given that it is foreign? 175/498 = 35.1%
9 PAGES 25 KEY Permutations and Combinations For 15, find the number of permutations , How many ways can you plant a rose bush, a lavender bush and a hydrangea bush in a row? 6 5. How many ways can you pick a president, a vice president, a secretary and a treasurer out of 28 people for student council? 491,400 For 610, find the probabilities. 6. What is the probability that a randomly generated arrangement of the letters A,E,L, Q and U will result in spelling the word EQUAL? What is the probability that a randomly generated 3letter arrangement of the letters in the word SPIN ends with the letter N? A bag contains ten chips numbered 0 through 9. Two chips are drawn randomly from the bag and laid down in the order they were drawn. What is the probability that the 2digit number formed is divisible by 3? 33/ 10 P 2 = 11/30 9. A prepaid telephone calling card comes with a randomly selected 4digit PIN, using the digits 1 through 9 without repeating any digits. What is the probability that the PIN for a card chosen at random does not contain the number 7? (8*7*6*5)/ 9 P 4 = 5/9 10. Janine makes a playlist of 8 songs and has her computer randomly shuffle them. If one song is by Little Bow Wow, what is the probability that this song will play first? 1/8 For 1113, calculate the number of combinations: For 1418, a town lottery requires players to choose three different numbers from the numbers 1 through How many different combinations are there? What is the probability that a player s numbers match all three numbers chosen by the computer? 1/ What is the probability that two of a player s numbers match the numbers chosen by the computer? 1/50,979, What is the probability that one of a player s numbers matches the numbers chosen by the computer? 1/ What is the probability that none of a player s numbers match the numbers chosen by the computer? 31/ Looking at the odds that you came up with in question 14, devise a sensible payout plan for the lottery in other words, how big should the prizes be for players who match 1, 2, or all 3 numbers? Assume that tickets cost $1. Don t forget to take into account the following: a. The town uses the lottery to raise money for schools and sports clubs. b. Selling tickets costs the town a certain amount of money. c. If payouts are too low, nobody will play! Answers will vary.
10 PAGES KEY Investigation: Theoretical vs. Experimental Probability Part 1: Theoretical Probability Probability is the chance or likelihood of an event occurring. We will study two types of probability, theoretical and experimental. Theoretical Probability: the probability of an event is the ratio or the number of favorable outcomes to the total possible outcomes. P(Event) = Number or favorable outcomes Total possible outcomes Sample Space: The set of all possible outcomes. For example, the sample space of tossing a coin is {Heads, Tails} because these are the only two possible outcomes. Theoretical probability is based on the set of all possible outcomes, or the sample space. 1. List the sample space for rolling a sixsided die (remember you are listing a set, so you should use brackets {} ): {1,2,3,4,5,6} Find the following probabilities: P(2) 1/6 P(3 or 6) 1/3 P(odd) 1/2 P(not a 4) 5/6 P(1,2,3,4,5, or 6) 1 P(8) 0 2. List the sample space for tossing two coins: {(H,H), (H,T), (T,H), (T,T)} Find the following probabilities: P(two heads) 1/4 P(one head and one tail) 1/2 P(head, then tail) 1/4 P(all tails) 1/4 P(no tails) 1/4 3. Complete the sample space for tossing two sixsided dice: {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} Find the following probabilities: P(a 1 and a 4) 1/18 P(a 1, then a 4) 1/36 P(sum of 8) 5/36 P(sum of 12) 1/36 P(doubles) 1/6 P(sum of 15) 0 4. When would you expect the probability of an event occurring to be 1, or 100%? Describe an event whose probability of occurring is 1. Any event that will definitely happen will have a probability of 1. Ex: rolling a 1,2,3,4,5,or 6 on die. 5. When would you expect the probability of an event occurring to be 0, or 0%? Describe an event whose probability of occurring is 0.
11 Any event that cannot happen will have a probability of 0. Ex: rolling an 8 on a sixsided die. Part 2: Experimental Probability Experimental Probability: the ratio of the number of times the event occurs to the total number of trials. P(Event) = Number or times the event occurs Total number of trials 1. Do you think that theoretical and experimental probabilities will be the same for a certain event occurring? Explain your answer. Answers will vary. At this point, some students may say that they will be the same. Actually, the experimental probabilities get closer to the theoretical probabilities as the number of trials increases. 2. Roll a sixsided die and record the number on the die. Repeat this 9 more times Number on Die Tally Frequency Total 10 Based on your data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) Answers will vary for the rest of this section. It will be based on the actual trials. How do these compare to the theoretical probabilities in Part 1? Why do you think they are the same or different? 3. Record your data on the board (number on die and frequency only). Compare your data with other groups in your class. Explain what you observe about your data compared to the other groups. Try to make at least two observations. 4. Combine the frequencies of all the groups in your class with your data and complete the following table: Number on Die Total Frequency Based on the whole class data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) How do these compare to your group s probabilities? How do these compare to the theoretical probabilities from Part 1? What do you think would happen to the experimental probabilities if there were 200 trials? 500 trials? 1000 trials? 1,000,000 trials? As the number of trials increases we should see the experimental probabilities approach the theoretical probabilities. 5. On your graphing calculator, go to APPS and open Prob Sim. Press any key and then select 2: Roll dice.
12 Click Roll. Notice that there will be a bar on the graph at the right. What does this represent? Now push +1 nine more times. Push the right arrow to see the frequency of each number on the die. How many times did you get a 1? A 2? A 5? Now press the +1, +10, and +50 buttons until you have rolled 100 times. Based on the data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) Press the +50 button until you have rolled 1000 times. Based on the data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) Press the +50 button until you have rolled 5000 times. Based on the data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) What can you expect to happen to the experimental probabilities in the long run? In other words, as the number of trials increases, what happens to the experimental probabilities? As the number of trials increases, the experimental probability should approach the theoretical probability Why can there be differences between experimental and theoretical probabilities in general? Theoretical probabilities tell us what we can expect to happen in the long run. Experimental probability is dependent on the number of trials conducted. Also, just because we know how often something should occur, that does not mean it actually will occur.
13 Part 3: Which one do I use? So when do we use theoretical probability or experimental probability? Theoretical probability is always the best choice, when it can be calculated. But sometimes it is not possible to calculate theoretical probabilities because we cannot possible know all of the possible outcomes. In these cases, experimental probability is appropriate. For example, if we wanted to calculate the probability of a student in the class having green as his or her favorite color, we could not use theoretical probability. We would have to collect data on the favorite colors of each member of the class and use experimental probability. Determine whether theoretical or experimental probability would be appropriate for each of the following. Explain your reasoning: 1. What is the probability of someone tripping on the stairs today between first and second periods? Experimental. We would need to collect data on the total number of students on the stairs between first and second period and how many of those students tripped. 2. What is the probability of rolling a 3 on a sixsided die, then tossing a coin and getting a head? Theoretical. We could list the sample space and find the probability based on the possible outcomes. 3. What is the probability that a student will get 4 of 5 true false questions correct on a quiz? Theoretical. We could list the sample space and find the probability based on the possible outcomes. 4. What is the probability that a student in class is wearing exactly four buttons on his or her clothing today? Experimental. We would have to ask each student how many buttons he or she is wearing to find the probability since we could not possible know all the possible outcomes.
14 PAGES KEY Probability Homework: Experimental vs. Theoretical Name 1) A baseball collector checked 350 cards in case on the shelf and found that 85 of them were damaged. Find the experimental probability of the cards being damaged. Show your work. 85/350 =.24 2) Jimmy rolls a number cube 30 times. He records that the number 6 was rolled 9 times. According to Jimmy's records, what is the experimental probability of rolling a 6? Show your work. 9/30 =.3 3) John, Phil, and Mike are going to a bowling match. Suppose the boys randomly sit in the 3 seats next to each other and one of the seats is next to an aisle. What is the probability that John will sit in the seat next to the aisle? 1/3 4) In Mrs. Johnson's class there are 12 boys and 16 girls. If Mrs. Johnson draws a name at random, what is the probability that the name will be that of a boy? 12/28 =.43 5) Antonia has 9 pairs of white socks and 7 pairs of black socks. Without looking, she pulls a black sock from the drawer. What is the probability that the next sock she pulls out will also be black? 13/31 =.42 (There were 32 socks to start with, but one black sock was removed. That leaves 31 socks, 13 of which are black.) 6) Lenny tosses a nickel 50 times. It lands heads up 32 times and tails 18 times. What is the experimental probability that the nickel lands tails? 18/50 =.36 7) A car manufacturer randomly selected 5,000 cars from their production line and found that 85 had some defects. If 100,000 cars are produced by this manufacturer, how many cars can be expected to have defects? 85/5000 =.017;.017*100,000 = 1700 cars can be expected to have defects. (Source: The following advertisement appeared in the Sunday paper:
15 Chew DentaGum! 4 out of 5 dentists surveyed agree that chewing DentaGum after eating reduces the risk of tooth decay! So enjoy a piece of delicious DentaGum and get fewer cavities! 10 dentists were surveyed. 8) According to the ad, what is the probability that a dentist chosen at random does not agree that chewing DentaGum after meals reduces the risk of tooth decay? 1/5 or.2 9) Is this probability theoretical or experimental? How do you know? Experimental because it is based on a survey or data collected. 10) Do you think that the this advertisement is trying to influence the consumer to buy DentaGum? Why or why not? Yes, the fine print states that only 100 dentists were surveyed. The results may be different if a larger sample was surveyed. 11) What could be done to make this advertisement more believable? The sample of dentists could be made larger. 10 dentists does not give a representative sample of all dentists. The larger the sample, the more accurate the probabilities will be.
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