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Chapter 4 Review MDM4U Jensen Section 1: Experimental Probability 1. Explain how you could conduct a simulation to determine the probability of guessing 20% of the questions correctly on a ten question multiple-‐choice test where each question has five possible answers. 2. A particular racehorse has an 85% success rate based on its winning races this year. Design and describe a simulation that will help you determine the experimental probability that the horse might lose two races in a row. 3. a) Use your graphing calculator to conduct a simulation of drawing a card from a standard deck of cards. The card is replaced after each time one is drawn. Complete a chart like the one below using your results to help determine the experimental probability of drawing an ace. Do 10 trials at a time and record the cumulative frequencies.
b) What is the simple event that is the focus of this experiment? c) Use your results to determine the experimental probability of drawing an ace.
Number of Cards Drawn
Total Number of Aces Drawn
10
20
30
40
50
60
70
80
90
100
Section 2: Theoretical Probability 4. Determine the theoretical probability for each of the following events. a) rolling a 6 on a die b) drawing a black card from a well-‐shuffled deck c) drawing a red 2 from a well-‐shuffled deck d) rolling a “Q” on a die with each side containing a letter of the English alphabet e) rolling a vowel 5. Each of the letters of the word PROBABILITY is printed on same-‐sized pieces of paper and placed in a bag. The bag is shaken and one piece of paper is drawn. (Consider Y as a vowel.) a) What is the probability that the letter A is selected? b) What is the probability that the letter B is selected? c) What is the probability that a vowel is selected? d) What is the probability that a consonant is not selected? 6. Eight students, five boys and three girls, must run a race in phys-‐ed class today. They all have an equal chance of winning the race. a) What is the probability that a girl wins the race? b) What is the probability that a girl does not win the race?
7. A lottery-‐mixing bin contains 149 balls numbered 1 to 149. If a winning ball is drawn at random, find the probability that the winning ball is between 10 and 20 inclusive. Section 3: Probability Using Sets 8. A magician’s hat contains three types of rabbits: floppy-‐eared, white, and baby rabbits. a) Draw a Venn diagram to illustrate the contents of the magician’s hat, given the following information: b) Suppose the magician selects one rabbit from the hat at random. Determine the probability that the magician chooses: i) a white rabbit ii) a rabbit without floppy ears iii) a baby rabbit without floppy ears.
In the hat # of Rabbits Floppy Eared 27
White 25
Baby 34
Floppy eared and white 13
Floppy eared baby 9
White baby 5
Floppy eared white baby 2
9. For each of the following, find the indicated probability and state whether A and B are mutually exclusive. a) P(A) = 0.2, P(B) = 0.4, P(A ∪ B) = 0.5, P(A ∩ B) = ? b) P(A) = 0.8, P(B) = 0.1, P(A ∪ B) = ?, P(A ∩ B) = 0.25 c) P(A) = ?, P(B) = 0.15, P(A ∪ B) = 0.75, P(A ∩ B) = 0 10. Rory applies for two jobs. The probability that she will get Job A is !
!" and the probability that she will
get Job B is !!. If the probability that she will get both jobs is !
!", what is the probability that she will get at
least one of the jobs? 11. The probability that it will snow today is 0.6 and the probability that it will snow tomorrow is 0.35. The probability that it will snow both days is 0.2. What is the probability that it will not snow today or tomorrow?
12. In Ms. Tanti’s geography class, • 60% of the students are Canadian • 28% of the students are of Chinese origin • 15% are Canadian and of Chinese origin. What is the chance that a student drawn at random would be: a) Canadian or of Chinese origin? b) neither Canadian nor of Chinese origin? c) of Chinese origin, but not Canadian? Section 4: Conditional Probability 13. Hanna surveyed her grade and summarized responses to the question, “ Do you like the food at the school cafeteria?” Find each of the following probabilities. a) P(likes the cafeteria food | student is female) b) P(student is male | no opinion about the cafeteria food)
14. A student is chosen at random in Kim’s school. If the probability that the student is taking math this semester is !"
!", the probability that the student is on the school’s soccer team is !
!"#, and the probability
that the student is doing both is !!"#, determine 𝑃 𝑡𝑎𝑘𝑖𝑛𝑔 𝑚𝑎𝑡ℎ 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑜𝑐𝑐𝑒𝑟 𝑡𝑒𝑎𝑚).
15. If a fair die is tossed and the outcome is even, find the probability that the outcome is a 4. 16. What is the probability of drawing two queens in a row (without replacement) from a well-‐shuffled deck of 52 playing cards? 17. In a bag of candy canes, there are 14 broken candies mixed in with 20 good ones. If Neha chooses three candies at random from the bag, what is the probability that all three are good?
Section 5: Multiplication of Independent Events 18. A test has three multiple-‐choice questions that offer five possible answers. a) Draw a tree diagram that shows all the possible multiple-‐choice combinations for the answers to the test. b) Determine the probability that a student will get two out of three questions correct by guessing on all of them. c) Determine the probability that a student will get one out of three questions correct by guessing on all of them.
19. The weather forecaster in Treeville incorrectly predicts the weather 15% of the time. What is the probability that the forecaster correctly forecasts the weather the next three days in a row? 20. A menu has three choices for salad, six main dishes, and four desserts. How many different meals are available if you select a salad, a main dish and a dessert? 21. A box contains a mixture of three types of sporting equipment: eight baseballs, five hockey helmets, and twelve badminton birdies. William randomly takes an item from the box, replaces it, and then takes a second item. He keeps the second item only if he got the same item as the first. Calculate the probability of each of the following. a) He will be able to keep a hockey helmet. b) He will be able to keep a baseball. c) He will not be able to keep a birdie. 22. A baseball player has an on-‐base percentage of 60%. Calculate the following probabilities. a) She will be on base in two out of three attempts. b) She will not get on base three times in a row.
c) She will get on base at least two out of three times she is at bat. d) She will get on base the first attempt and then not get on base the next two. Section 6: Permutations 23. Evaluate each of the following a) 7! b) 8P2 c) 4! × 3! d) P(14, 5) e) !"!
!!!!!!
24. Simplify a) !!
!!! ! b) !!! !
!!!!! !
25. a) In how many ways can 9 students standing in a straight line be arranged? b) In how many ways can 9 students standing in a straight line be arranged if Jill must be first? 26. A combination lock opens when the right combination of three numbers from 00 to 59 is entered. The same number may not be used more than once. a) What is the probability of getting the correct combination by chance? b) What is the probability of getting the right combination if you already know the first digit? 27. The Pittsburgh Penguins have 12 forwards on their roster. In how many ways can they finish first, second, and third in scoring on their team? 28. How many distinct arrangements of the letters in PERMUTATIONS can you make? a) taking all the letters at a time b) taking seven letters at a time c) taking three letters at a time
29. Ten beads of different colours are strung on a string to make a necklace. How many different arrangements are there? 30. How many distinct four-‐digit odd numbers can be formed from the digits in the number 6 738 195? 31. Manpreet has 6 romance novels, 4 fiction novels, and 9 war novels on a shelf. Show a formula to calculate how many ways can she arrange her novels on the shelf if novels of the same genre are to be kept together? 32. A drawer contains five blue, seven pink and three orange scarves. Three scarves are drawn from the drawer, one at a time, without replacement. Determine the probability that the order in which they are selected is blue, pink, orange Section 7: Combinations 33. Evaluate each of the following a) C(8, 3) b) 2C1 c) !"! d) C(21, 13) e) 5C2
34. Solve for 𝑛 in the equation: C(n, 2) = 66 35. In how many ways can a child select three different ice cream flavours from nine different ice cream flavours? 36. How many hands of six cards can be selected from a standard deck of 52 cards? 37. In how many ways can a group of five people be chosen from seven couples (each of which has one male and one female) to form a club, given each of the following conditions? a) All are equally eligible for the club. b) The club must include two females and three males. 38. Two teachers and six students on a class trip must ride in two four-‐passenger cars. a) What is the number of ways that the eight people can be divided into two groups to ride in the two cars? b) What is the number of ways if only teachers are allowed to drive? c) What is the probability that the teachers will ride in the same car?
39. Monique has eight red jelly beans and six purple jelly beans in a jar. She pulls out one jelly bean. a) What are the odds in favour of the jelly bean being a red one? b) What are the odds against the jelly bean being a purple one? 40. The weather forecaster predicts that the probability of sun tomorrow is 60%. What are the odds in favour of sun tomorrow? Explain your answer. Answers 1. Write correct on one slip of paper and incorrect on four more. Draw one. Do this 10 times and see if you got 2 correct and 8 incorrect (20%). 2. Put 17 red chips and 3 blue chips in a hat. Draw 2 chips randomly, with replacement after the first draw; do this 100 times. How many times were 2 blue chips chosen? OR use randint (1, 100, 2). Do this 100 times. Record how many times both numbers were between 86 and 100. 3. a) use randint(1, 52, 10) and record the number of 1,2,3, or 4’s that are present. b) drawing an ace c) answers may vary 4. a) !
! b) 0.5 c) !
!"= !
!" d) !
!" e) !
!" (if the vowels are a, e, i, o, u
5. a) !
!! b) !
!! c) !
!! d) !
!!
6. a) !
! b) !
!
7. !!
!"#
8. a) b) i) !"!" ii) !"
!" iii) !"
!"
9. a) 0.1, no b) 0.65, no c) 0.6, yes 10. !!
!"
11. 0.25 12. a) 0.73 b) 27% c) 13% 13. a) !
! b) !"
!"
14. !"
!"
15. !
!
16. !
!!"
17. 0.19 18. a) b) !"
!"# c) !"
!"#
19. 0.614 20. 72 21. a) !
!" b) !"
!"# c) !"#
!"#
22. a) 0.432 b) 0.064 c) 0.648 d) 0.096
23. a) 5040 b) 56 c) 144 d) 240 240 e) 924 24. a) 𝑟 b) !
(!!!!!)
25. a) 362880 b) 40 320 26. a) !
!"#$!" b) !
!"##
27. 1320 28. a) 239 500 800 b) 1 995 840 c) 660 29. 1 814 400 30. 600 31. 6! × 4! × 9! × 3 × 2 32. 0.038 33. a) 56 b) 2 c) 210 d) 203 490 e) 10 34. 12 35. 84 36. 20 358 520 37. a) 2002 b) 735 38. a) 70 b) 40 c) !
!
39. a) 4:3 b) 4:3 40. 3:2