c Dr. Patrice Poage, March 1, 20 1 Math 1324 Review Questions for Test 2 (by Poage) covers sections 8.3, 8.4, 8.5, 9.1, 9.2, 9.3, 9.4 1. A basketball player has a 75% chance of making a free throw. What is the probability of her making 1 or more free throws in 120 trials. Use the normal curve approximation to the binomial distribution. 2. A box contains 7 toothpicks, 240 of which are cracked. If you randomly select 150 toothpicks, what is the probability at least 1 is cracked? JUST SET UP THE PROBLEM 3. A drawer contains 4 black-ink pens, 3 blue-ink pens, and 3 red-ink pens. You randomly reach in and select one pen out at a time (without replacement) until you have 2 black-ink pens. Let X be the random variable denoting the total number of pens you draw out. What values may X assume? 4. An English class had an exam with an average of 68.35 and a standard deviation of 14.76. If 6% of the class made A s, 32% of the class made B s, and the rest of the class made C s, what was the cutoff grade to make a B or higher? 5. Let Z be the standard normal random variable. Find a if: (a) P (Z <a)=0.234 (b) P ( a <Z<a)=0.286 6. Mason has a bucket of sidewalk chalk. In the bucket there are 2 blue, 8 green, 2 pink, 4 yellow, and 5 pieces of white chalk. If he randomly pulls out 6 pieces of chalk, in how many ways can he pull out exactly 2 green chalks and 1 white chalk? 7. An exam consists of four true/false questions. (a) In how many different ways can a person complete this exam if every question is answered? (b) What is the probability distribution for the number of correct answers? 8. Firestarters Inc. manufactures artificial starter logs for fireplaces. These logs are accepted by the buyer if they fall within the tolerance limits of.695 inches and.780 inches in length. Assuming that the length of the logs is normally distributed with a mean of.72 inches and a standard deviation of.03 inches, estimate the percentage of logs that will be rejected by the buyer. 9. There are 12 multiple choice questions on an exam in which each question has 5 answers. If Richard knows the answer to 8 of them, and randomly guesses at the remaining 4, what is the probability Richard will answer all 12 questions correctly? 10. A customer at a CD store selects four CD s from a stack of twenty-five in which six are scratched. What is the probability the customer selects at least three CD s that are NOT scratched?. Use the data to the right to find the following: mean standard deviation
c Dr. Patrice Poage, March 1, 20 2 variance median mode X P(X) 3 5 8 14.15.34.26.25 12. A bag contains 4 grape, 8 cherry, and 10 lime Jolly Ranchers. If Bob randomly selects 7 Jolly Ranchers from the bag, what is the probability he selects exactly 5 of the same color? 13. Classify the following random variables as FINITE DISCRETE, INFINITE DISCRETE, or CONTINUOUS. Give the values of the random variables. (a) A bag contains 3 red, 6 blue, and 4 white marbles. Marbles are drawn one at a time without replacement until a red one is drawn. Let X denote the number of marbles drawn in one trial of this experiment. (b) Let X be the temperature, in degrees Celsius, of a cup of hot chocolate. (c) Let X denote the number of minutes a person waits (in one particular day) in line to pull football tickets. (d) Let X denote the number of times my dog, Rebound, has to bark before I let him in my house. (e) Let X denote the number of cards that are drawn from a standard deck of cards, without replacement, until a King is drawn. (f) Let X be the number of times you roll a dice until a 3 appears. 14. A game costs $2. to play. This game consists of rolling a 10-sided die 1 time. If the die lands on a 9, you win $5. If it lands on a 1 or 8, you win $3. If it lands on a 2, 3, or 4, you win $1. For any other result, you lose. Let X denote the net winnings of a person playing this game. (a) Find the probability distribution for X. (b) How much can a person expect to win/lose if they play this game? 15. At a school cafeteria, 4 students ate bad sausage. The probability of getting food poison from bad sausage is 30%. (a) What is the probability at most 1 students get sick? (b) What is the probability at least 1 students get sick? (c) What is the probability that between 120 and 180 students get sick? (inclusive) (d) How many students can you expect to get sick? Grade Frequency A B C D F 9 18 24 6 16. The table above shows the all the grades for Mr. Teacher s class over the last three semesters. (a) Find the probability distribution for this data. (b) If a student takes this class, what is the probability that the student passes (assuming D s and F s are failing)?
c Dr. Patrice Poage, March 1, 20 3 17. The manager of Poage Finance has estimated that, because of a recession year, 5% of its 4 loan accounts will be delinquent. If the manager s estimate is correct, use the Normal Approximation to the Binomial Distribution to find the probability that 25 or more of the accounts will be delinquent. 18. The weight of a length of rope will support is normally distributed with a mean of 20 lbs and a standard deviation of 50 lbs. What is the probability that you will buy a rope that can support between 19 lbs and 2050 lbs? 19. Four boys and five girls are randomly assigned seats in a row. What is the probability that all the girls sit together and all the boys sit together? 20. Below is a histogram for the random variable, X. Answer the next 5 questions using this data. (a) How high should the bar above the 6 be drawn? (b) What is the mode? (c) What is the E(X)? (d) What is the variance? (e) What is the median? (f) What is the standard deviation? Probability 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 3 4 5 6 7 8 9 X 21. In a group of 30 ballpoint pens on a shelf in the stationery department of the Metro Department Store, 2 are known to be defective. If a customer selects 3 of these pens, what is the probability that at least 1 is defective? 22. An engineering class takes an exam in which the average was 65.7 and the standard deviation 12.93. The prof decides to give F s to 8% of the class, D s to 18% of the class, B s to 25% of the class, A s to 10% of the class, and C s to the rest. What is the lowest grade you can make and still pass (assuming C or higher is passing). 23. It costs $2 to play a game which consists of drawing 3 toothpicks from a bucket containing 8 red, 6 blue, and 2 green toothpicks. If all 3 toothpicks are the same color, you win $5. If all 3 are different colors, you win $4. For everything else you lose. Let the random variable, X, denote your net winnings. (a) Find the probability distribution for X. (b) How much money can you expect to win/lose? 24. The probability that a man will be color blind is 0.038. What is the probability that in a group of 47 men, at most 2 are color-blind?
c Dr. Patrice Poage, March 1, 20 4 25. Toss a fair coin 20 times. What is the probability of getting: (a) exactly 4 heads (b) more than 10 heads (c) at most 15 heads (d) between 12 and 18 heads 26. A box of 24 Crayons contains three broken ones. If Julie selects 8 Crayons at random without replacement, what is the probability that she ll get at exactly 1 broken crayon? 27. Let Z be the standard normal random variable. Find P (Z 0.75). 28. Bob sells magazine subscriptions over the phone. He estimates that the probability of his making a sale with each attempt is 0.12. What is the probability of Bob making more than 10 sales if he makes 80 calls? 29. The length of a standard piece of plywood is normally distributed with a mean of 12 feet and a standard deviation of 9 inches. What is the probability that a piece of plywood is more than 10 feet long? 30. A barrel contains 5 green apples and 10 red apples. An experiment consists of pulling one apple out at a time, without replacement, until you have 3 red apples. Let the random variable X denote the number of apples pulled out of the barrel. What values may X assume? 31. At a factory, the boss has decided to evaluate his employees on a 1 point scale. The average score made was 76 with a standard deviation of 12.3. He decides to assign a rating of EXCEL- LENT to the top 37%, a rating of SATISFACTORY to the next 43%, and a rating of POOR to the rest. What is the lowest score an employee could get and still be in the SATISFACTORY category? Assume the scores are normally distributed. 32. A weighted coin (the probability it lands on heads is 0.72) is flipped 5 times. What is the probability of the coin landing on TAILS at least 30 times? 33. The data below shows the scores received on the AP Calculus test by students at Lovelady High School last year. Use the data to answer the following 3 questions. (a) What is the standard deviation? (b) What is the median? Score Freq 0 1 2 3 4 5 4 15 26 19 12 8 (c) What is the P (X >2)? 34. A marksman s chance of hitting a target with each of his shots is 60%. If he firest 30 shots, use the Normal curve approximation to the Binomial distribution to find the probability he hits at least 15, but no more than 20 times. 35. A game consists of rolling a pair of fair dice. If the sum is 2 or 12, you win $13. If the sum is 7 or, you win $8. For everything else you lose. If it costs $3 to play this game, what are the expected net winnings of one play? Interpret your answer.
c Dr. Patrice Poage, March 1, 20 5 36. At a candy factory, 12% of all peppermints produced leave the factory with no wrapper. Yoda randomly selects 4 pieces peppermints from this factory. Let X denote the random variable of the number of peppermints selected that have no wrapper. Find the probability distribution for this data. 37. A set of 2 animals is to be sold from store that has of 5 cats, 3 dogs, and 4 birds. Let X be the random variable denoting the number of birds in the set. Find the probability distribution for X. Write probabilities as reduced fractions. 38. The life span of a 60 watt light bulb is normally distributed with an average life span of 8,0 hours and a standard deviation of 15 days. What is the probability that a bulb selected at random will last at least 8,250 hours? 39. Use the histogram to answer the next 6 questions: (a) Find the standard deviation. (b) Find the median. (c) Find the mode. (d) Find the mean. (e) Find the variance. (f) Find P (X <4).35.30.25.20.15.10.05 1 2 3 4 5 6 7 8 40. The scores on a History exam are normally distributed with a mean of 65 and a standard deviation of 12. If the instructor assigns A s to 24% of the class, what is the minimum grade a student can make on the exam and still receive an A on the test? 41. Let Z be the standard normal random variable. Find the value of a if P ( a Z a) =0.6842. 42. A bag contains 7 blue balls, 9 red balls, and 10 yellow balls. In how many ways can a sample of 8 balls be selected if at least 2 of them are to be yellow? 43. Cheryl planted three dozen flowers in her front flower bed. If each flower has an 80% change of blooming, how many flowers can she expect to bloom? 44. An exam consists of 10 multiple choice questions, each with 5 choices. If Mark knows the answer to 4 of them, and guesses and the rest, what is the probability he will get at least 7 of them correct?
c Dr. Patrice Poage, March 1, 20 6 45. An automobile manufacturer receives the microprocessors used to regulate fuel consumption in its automobiles in shipments of 10 each from a certain supplier. It has been estimated that, on the average, 1% of the microprocessors manufactured by the supplier are defective. Use the normal curve approximation to the binomial distribution to find the probability that more than 12 of the microprocessors in a single shipment are defective. 46. Texas PB&J University (a VERY small university) estimates that 91.5% of the freshman class will graduate within four years. From the incoming class of 60 students, what is the probability at least 52 will graduate within four years? 47. Three cards are drawn from a standard deck of 52. Let X be the number of Aces drawn. Find P (X =2). 48. Suppose you pay $5 to play a game where you toss 3 coins. You receive $1 if one head results, $4 if 2 heads result, and $9 if three heads result. What are your expected net winnings? 49. A new drug cures 70% of the people taking it. Suppose 20 people take the drug; find the probability of: (a) exactly 18 people being cured (b) at least 10, but no more than 16 are cured. 50. Classify the random variables as finite discrete, infinite discrete, or continuous and give the values of X. (a) X= the # of minutes you spend studying the day before the test (b) X = the number of jokes your math prof tells until you laugh (c) X = the number of times it takes you to draw out 3 red marbles in a box containing 5 red and 7 yellow marbles. (d) X = the number of cards drawn from a deck of 52, without replacement, until a King is drawn. 51. Let Z be the standard normal random variable. Find the value of a if (a) P (Z >a)=0.75 (b) P (Z <a)=0.96 (c) P ( a <Z<a)=0.426 52. Two light bulbs are selected at random from a box of 24, of which 4 are defective. What is the probability at most 1 is defective? 53. A box contains six red, five black, and four green balls. If three are selected at random, what is the probability all 3 are the same color?