MATHEMATICS 20-BNJ-05 Topics in Mathematics Martin Huard Winter 204 XXII Probability. Find the sample space S along with n S. a) The face cards are removed from a regular deck and then card is selected from this set of 2 face cards. A box contains three marbles of different colors, blue, red and green. Two balls are drawn at random (without replacement). A box contains three marbles of different colors, blue, red and green. Two balls are drawn at random (with replacement). d) Two dice are rolled, and the sum of their dots is observed. e) Two dice are rolled, and the product of their dots is observed. f) Three coins are tossed and the outcome is recorded. 2. A single die is rolled. Find the probability the number on top is: a) a 3. an odd number. a number less than 5. d) a number no more than 3. 3. Two dice are rolled, where one is black and the other is white. Find the following probabilities. a) P(white die is an odd number) P(sum is 6) P(both dice show odd numbers) d) P(# on black die > # on white die) 4. The odds of being accepted in Mathematics at McGill University are 3 to 8. Find the probability of being accepted. 5. A box contains 3 marbles, a red, a blue and a green marble. In two marbles are picked at random, what is the probability that a) one is red and the other is green? both are red? 6. A card is drawn from a standard deck of 52 cards. Find the following probabilities. a) P( King ) P( club ) P( black card) d) P( Face card ) 7. A group of 50 randomly selected CEOs was tested for personality type. The following table gives the results of this survey. Type A Type B Men 78 42 Women If one CEO is selected at random from this group, find the probability that this CEO a) has a type A personality is a women is a women with a type A personality
8. The academic adviser of a college informs us that in his college of 4000 students, 500 students are currently enrolled in a philosophy class and 550 in a mathematics class. If 220 students are in a philosophy and a mathematics class, find the probability that if a student is selected at random, he will be in: a) only a philosophy class only a mathematics class a least one of the two classes. 9. The newsletter of a College announced that 75 students are in a math or poetry club. If the math club has 92 members, with 35 also in the poetry club, what is the probability that if a student is picked at random a) he will be in the math club only. he will be in the poetry club only. he will be in both clubs. 0. A group of 90 freshman engineering students at a large university was surveyed with the following results. of the students read Scientific American 8 read Popular Mechanics 50 read Mathematics Magazine 3 read Scientific American and Popular Mechanics read Popular Mechanics and Mathematics Magazine 3 read Scientific American and Mathematics Magazine 9 read all three Using this data, determine the probability that if a student is chosen at random, he will read a) none of the publications? Mathematics Magazine only? How many read Scientific American and Popular Mechanics, but not Mathematics Magazine?. A survey on a group of 00 College students showed that 8 of them have a motorcycle, 20 have a car, 48 have a bicycle and 38 have neither a motorcycle, a car or a bicycle. No one has at the same time a motorcycle and a car. What is the probability that a student chosen at random has a bicycle and either a car or a motorcycle? 2. Two cards are drawn from a deck of 52 cards. What is the probability that a) both are clubs both are Jacks. 3. A single card is drawn from a deck of 52 cards. What is the probability that the card is a) a king of diamonds? a king or a diamond? a face card? d) a red face card? e) not an ace? 4. An urn contains 20 marbles, of which 5 are red, 6 are blue, 7 are yellow and 2 are green. If two marbles are chosen at random, what is the probability that a) both are blue? one is blue and the other is red? both are of the same color? d)the two marbles are not the same color? Winter 204 Martin Huard 2
5. In a refugee camp in Rwanda, it was found that 90% of the refugees came to escape political oppression, 80% came to escape abject poverty, and 70% came to escape both. What is the probability that a refugee in the camp was neither poor nor seeking political asylum? 6. A survey conducted about job satisfaction showed that 20% of workers are not happy with their current job. Assume that this result is true for the population of all workers. Two workers are selected at random, and it is observed whether or not they are happy with their current jobs. Find the probability that in this sample of two workers, a) both are not happy with their current jobs. at least one of them is happy with the current job. 7. Suppose a birth control pill is 99% effective in preventing pregnancy. a) What is the probability that none of 00 women using the pill will become pregnant? What is the probability that at least one woman per 00 users will become pregnant? 8. A company has installed a generator to back up the power in case there is a power failure. The probability that there will be a power failure during a snowstorm is 0.30. The probability that the generator will stop working during a snowstorm is 0.09. What is the probability that during a snowstorm the company will lose both sources of power?. An old age home employs 65 people. Eight of the 30 men and 2 of the 35 women are nurses. What is the probability that an employee picked at random is a man or is a nurse? 20. Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two-way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same, or worse off than their parents. Education Level High school or CEGEP More than less CEGEP Better off 40 450 420 Same 60 250 0 Worse off 200 300 70 Suppose one adult is selected at random from these 2000 adults. Find the following. a) P(better off or CEGEP) P(More that CEGEP or worse off) P(better off or worse off) d) P(better off given CEGEP) e) Are the events better off and CEGEP independent? f) Are the events better off and CEGEP mutually exclusive? 2. What is the probability that in a group of 4 students, no two students have their birthday in the same month? 22. If a committee of five is chosen from a pool of 2 candidates, what is the probability that two of the candidates, Galois and Abel, will not sit on the committee together? Winter 204 Martin Huard 3
23. In a group of 2 SLC students, 5 are in science, 4 in social science, and 3 in business. If a group of four is chosen, what is the probability that the group a) Will consist of only science students? Will have at least one science student? At least one student from each program? 24. A coin is tossed eight times. Find the probability of having a) No Heads At least one Head Exactly two heads d) At least two Heads 25. The odds that Robin will hit a target with his arrow is 4 to. If he shoots five arrows, what is the probability that a) he will hit the target only with his fifth arrow? He will hit the target at least once? He will hit the target with at least four arrows? d) He will hit the target with exactly three arrows? 26. It is estimated that 5% of a large consignment of eggs in a certain supermarket are broken. a) What is the probability that a customer who randomly selects a dozen of these eggs receives at least one broken egg? What is the probability that a customer who selects these eggs at random will have to check more than three cartons before finding a carton without any broken eggs? (Each carton contains a dozen eggs). 27. Alex, Bill and Joe each in turn toss a balanced coin. The first one to throw a head wins. a) What are their respective chances of winning if each tosses only one time? What are their respective chances of winning if they continue, when there is no winner, giving a maximum of two tosses each? 28. The probability that a certain door is locked is 0.6. The key to the door is one of five unidentified keys hanging of a key rack. Two keys are randomly selected before approaching the door. What is the probability that the door may be opened without returning for another key? 29. At SLC, 60% of students in science are in the Pure and Applied profile. Of these, 95% love mathematics, while 75% of students in the Health profile love mathematics. Find the probability that a science student chosen at random a) loves math and is in the Health profile loves math 30. A smooth-talking young man has a /3 probability of talking a policeman out of giving him a speeding ticket. The probability that he is stopped for speeding during a given weekend is ½. Find the probability that he will receive no speeding tickets a) on a given weekend? on 3 consecutive weekends? 3. Five black balls and four white balls are placed in an urn. Two balls are then drawn in succession. What is the probability that the second ball drawn is a white ball a) if the second ball is drawn without replacing the first? If the first ball is replaced before the second is drawn? Winter 204 Martin Huard 4
32. Two cards are drawn in succession without replacement of a deck of 52 cards. a) What is the probability that the first card is a heart given that the second card is a heart? What is the probability that the first card is a heart given the second card is a diamond? What is the probability that the first card is a jack given that the second card is an ace? 33. Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. a) What is the probability that the transferred ball was white given that the second ball drawn was white? What is the probability that the transferred ball was black given that the second ball drawn was white? 34. As accounts manager in your company, you classify 75% of your customers as "good credit" and the rest as "risky credit" depending on their credit rating. Customers in the "risky" category allow their accounts to go overdue 55% of the time on average, whereas those in the "good" category allow their accounts to become overdue only 0% of the time. What percentage of overdue accounts is held by customers in the "risky credit" category? 35. A witness sees a crime involving a taxi in Cabcity. The witness says that the taxi is green. It is known from previous research that witnesses are correct 80% of the time when making such statements. The police also know that 75% of the taxis in Cabcity are green, the other 25% being blue. What is the probability that a green taxi was involved in the crime? 36. A car insurance company has compiled the following table concerning the age distribution of its clients. Age % of clients % of group who are made a claim last year 25 and under 5 30 26-64 60 50 65 and over 25 25 a) What is the probability that a client selected at random made a claim last year? If a client made a claim last year, what is the probability that he or she is in the 25 and under age bracket? 37. Suppose the weather in a certain city is either rainy or sunny. Past records show that if it is sunny today, then it will be sunny tomorrow with a probability of 0.8 and if it is rainy today, then it will be rainy tomorrow with a probability of 0.7. Today is a sunny day. a) What is the probability it will be sunny n days from now? What proportion of days are sunny? 38. Sheldon either reads a math book or a physics book. If reads a math book, the probability the next book will be a math one is ¾, and if he reads a physics book, the probability the next one will be a math one is ½. a) What is the probability he will read a math book n books from now if he is currently reading a physics book. What proportion of books read are math books? Winter 204 Martin Huard 5
ANSWERS. a) S = { J, J, J, J, Q, Q, Q, Q, K, K, K, K } n S 2 S BR, BG, RB, RG, GB, GR n S 6 S BB, BR, BG, RB, RR, RG, GB, GR, GG n S 9 d) S 2,3, 4,5,6,7,8,9,0,,2 n S e) S, 2,3, 4,5,6,8,9,0,2,5,6,8, 20, 24, 25,30,36 n S 8 f) S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT n S 8 2. a) 6 2 2 3 d) 2 3. a) 2 5 36 4 d) 5 2 3 4. 5. a) 3 0 6. a) 3 4 2 d) 3 3 7. a) 97 50 5 50 8. a) 8 25 9. a) 57 75. 7 50 83 75 5 3. a) 52 4 3 3 3 d) 3 26 4. a) 3 38 3 47 0 43 d) 0 5. 0 6. a) 25 99 7. a) 00 99 00 0.366 00 8. 27 000 20. a) 39 50 2. 55 96 20. 5 65 0.6 4 00 3 79 00 d) 9 20 33 400 83 400 3 0. a) 90 7 8 2 45 2. a) 7 22 e) 2 3 24 25 e) No since P(better off) = 0 P(better off given CEGEP) = 9 200 20 f) No since P(better off and CEGEP) = 9 40 0 24. a) 256 22. 28 33 255 256 7 64 26. a) 2 0.460 20 23. a) 99 247 d) 25. a) 4 256 325 2 3 20 0.097 27. a) A = 2 B = 4 J = 8 A = 9 28. 6 25 9 6 32 92 99 6 324 325 9 B C 29. a) 3 0 87 00 30. a) 2 3 8 27 3. a) 4 9 4 9 32. a) 4 7 3 5 4 5 8 33. a) 7 9 7 34. 7 36. a) 63 400 8 38. a) P M n 35. 2 3 2 3 63 37. a) 2 n 2 3 4 3 2 3 64 2304 325 n P Sn 5 2 5 3 5 d) 28 625 Winter 204 Martin Huard 6