DISCRETE VARIABLE PROBLEMS ONLY

Transcription

1 DISCRETE VARIABLE PROBLEMS ONLY. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters the player receives in return for each score. Score Probability Number of counters player receives n Find the value of n in order for the player to get an expected return of 9 counters per roll. [4] 2. The quality control department of a company making computer chips knows that 2% of the chips are defective. Use the normal approximation to the binomial probability distribution, with a continuity correction, to find the probability that, in a batch containing 000 chips, between 20 and 30 chips (inclusive) are defective. [7] 3. A supplier of copper wire looks for flaws before despatching it to customers. It is known that the number of flaws follow a Poisson probability distribution with a mean of 2.3 flaws per metre. Determine the probability that there are exactly 2 flaws in metre of the wire. Determine the probability that there is at least one flaw in 2 metres of the wire. [3] 4. The continuous random variable X has probability density function f (x) where f k (x) = e 0, ke kx, 0 x otherwise Show that k =. What is the probability that the random variable X has a value that lies between and? Give your answer in terms of e. 4 2 Find the mean and variance of the distribution. Give your answers exactly, in terms of e. The random variable X above represents the lifetime, in years, of a certain type of battery. (d) Find the probability that a battery lasts more than six months. A calculator is fitted with three of these batteries. Each battery fails independently of the other two. Find the probability that at the end of six months (e) none of the batteries has failed;

2 (f) exactly one of the batteries has failed. [7] 5. In a game a player rolls a biased tetrahedral (four-faced) die. The probability of each possible score is shown below. Score Probability x Find the probability of a total score of six after two rolls. [3] 6. The probability distribution of a discrete random variable X is given by x P(X = x) = k 2, for x = 0,, 2,... 3 Find the value of k. [3] 7. A satellite relies on solar cells for its power and will operate provided that at least one of the cells is working. Cells fail independently of each other, and the probability that an individual cell fails within one year is 0.8. For a satellite with ten solar cells, find the probability that all ten cells fail within one year. For a satellite with ten solar cells, find the probability that the satellite is still operating at the end of one year. For a satellite with n solar cells, write down the probability that the satellite is still operating at the end of one year. Hence, find the smallest number of solar cells required so that the probability of the satellite still operating at the end of one year is at least [9] 8. In a school, of the students travel to school by bus. Five students are chosen at random. Find the probability that exactly 3 of them travel to school by bus. [3] 9. X is a binomial random variable, where the number of trials is 5 and the probability of success of each trial is p. Find the values of p if P(X = 4) = 0.2. [3] 0. Patients arrive at random at an emergency room in a hospital at the rate of 5 per hour throughout the day. Find the probability that 6 patients will arrive at the emergency room between 08:00 and 08:5. The emergency room switchboard has two operators. One operator answers calls for doctors and the other deals with enquiries about patients. The first operator fails to answer % of her calls and the second operator fails to answer 3% of his calls. On a typical day, the first and second telephone operators receive 20 and 40 calls respectively during an afternoon session. Using the Poisson distribution find the probability that, between them, the two operators fail to answer two or more calls during an afternoon session. [8] 2

3 2. A coin is biased so that when it is tossed the probability of obtaining heads is. The coin is 3 tossed 800 times. Let X be the number of heads obtained. Find the mean of X; the standard deviation of X. [3] 2. When John throws a stone at a target, the probability that he hits the target is 0.4. He throws a stone 6 times. Find the probability that he hits the target exactly 4 times. Find the probability that he hits the target for the first time on his third throw. 3. Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The score for each child is the sum of the two numbers shown on their respective dice. (i) Calculate the probability that Alan obtains a score of 9. (ii) Calculate the probability that Alan and Belle both obtain a score of 9. (i) (ii) Calculate the probability that Alan and Belle obtain the same score, Deduce the probability that Alan s score exceeds Belle s score. Let X denote the largest number shown 4 on the four dice. x (i) Show that for P(X x) = 6, for x =, 2,... 6 (ii) Copy and complete the following probability distribution table. x P(X = x) (iii) Calculate E(X). [3] 4. The random variable X is Poisson distributed with mean and satisfies P(X = 3) = P(X = 0) + P(X = ). Find the value of, correct to four decimal places. For this value of evaluate P(2 X 4). 5. Give your answers to four significant figures. A machine produces cloth with some minor faults. The number of faults per metre is a random variable following a Poisson distribution with a mean 3. Calculate the probability that a metre of the cloth contains five or more faults. [4] 3

4 6. When a boy plays a game at a fair, the probability that he wins a prize is He plays the game 0 times. Let X denote the total number of prizes that he wins. Assuming that the games are independent, find E(X) P(X 2). 7. Two typists were given a series of tests to complete. On average, Mr Brown made 2.7 mistakes per test while Mr Smith made 2.5 mistakes per test. Assume that the number of mistakes made by any typist follows a Poisson distribution. Calculate the probability that, in a particular test, (i) (ii) (iii) Mr Brown made two mistakes; Mr Smith made three mistakes; Mr Brown made two mistakes and Mr Smith made three mistakes. In another test, Mr Brown and Mr Smith made a combined total of five mistakes. Calculate the probability that Mr Brown made fewer mistakes than Mr Smith. [] 8. On a television channel the news is shown at the same time each day. The probability that Alice watches the news on a given day is 0.4. Calculate the probability that on five consecutive days, she watches the news on at most three days. 9. The random variable X has a Poisson distribution with mean λ. Given that P(X = 4) = P(X = 2) + P(X = 3), find the value of λ. Given that λ = 3.2, find the value of (i) P(X 2); (ii) P(X 3 X 2). [8] 20. The random variable X has a Poisson distribution with mean λ. Let p be the probability that X takes the value or 2. Write down an expression for p. Sketch the graph of p for 0 λ 4. Find the exact value of λ for which p is a maximum. [7] 2. Let X be a random variable with a Poisson distribution such that Var(X) = (E(X)) 2 6. Show that the mean of the distribution is 3. Find P(X 3). Let Y be another random variable, independent of X, with a Poisson distribution such that 4

5 E(Y) = 2. Find P(X + Y < 4). (d) Let U = X + 2Y. (i) Find the mean and variance of U. (ii) State with a reason whether or not U has a Poisson distribution. [0] 22. A discrete random variable X has its probability distribution given by P(X = x) = k(x + ), where x is 0,, 2, 3, 4. Show that k = 5 Find E(X). 23. The random variable X has a Poisson distribution with mean 4. Calculate P(3 X 5); P(X 3); P(3 X < 5X 3). 24. Consider the 0 data items x, x 2,... x 0. Given that = 34 and the standard deviation is 6.9, find the value of x. 0 i 2 x i 25. Let X be a random variable with a Poisson distribution, such that P(X > 2) = Find P(X < 2). [4] 26. The number of car accidents occurring per day on a highway follows a Poisson distribution with mean.5. Find the probability that more than two accidents will occur on a given Monday. Given that at least one accident occurs on another day, find the probability that more than two accidents occur on that day. 27. Andrew shoots 20 arrows at a target. He has a probability of 0.3 of hitting the target. All shots are independent of each other. Let X denote the number of arrows hitting the target. Find the mean and standard deviation of X. Find (i) P(X = 5); (ii) P(4 X 8). 5

6 Bill also shoots arrows at a target, with probability of 0.3 of hitting the target. All shots are independent of each other. Calculate the probability that Bill hits the target for the first time on his third shot. (d) Calculate the minimum number of shots required for the probability of at least one shot hitting the target to exceed [9] 28. The random variable X follows a Poisson distribution. Given that P(X ) = 0.2, find the mean of the distribution; P(X 2). 29. A bag contains a very large number of ribbons. One quarter of the ribbons are yellow and the rest are blue. Ten ribbons are selected at random from the bag. Find the expected number of yellow ribbons selected. Find the probability that exactly six of these ribbons are yellow. Find the probability that at least two of these ribbons are yellow. (d) Find the most likely number of yellow ribbons selected. (e) What assumption have you made about the probability of selecting a yellow ribbon? [2] 30. In an experiment, a trial is repeated n times. The trials are independent and the probability p of success in each trial is constant. Let X be the number of successes in the n trials. The mean of X is 0.4 and the standard deviation is 0.6. Find p. Find n. 3. A biology test consists of seven multiple choice questions. Each question has five possible answers, only one of which is correct. At least four correct answers are required to pass the test. Juan does not know the answer to any of the questions so, for each question, he selects the answer at random. Find the probability that Juan answers exactly four questions correctly. Find the probability that Juan passes the biology test. 32. The time, T minutes, spent each day by students in Amy s school sending text messages may be modelled by a normal distribution. 30 of the students spend less than 0 minutes per day. 35 spend more than 5 minutes per day. Find the mean and standard deviation of T. The number of text messages received by Amy during a fixed time interval may be modelled by a 6

7 Poisson distribution with a mean of 6 messages per hour. Find the probability that Amy will receive exactly 8 messages between 6:00 and 8:00 on a random day. Given that Amy has received at least 0 messages between 6:00 and 8:00 on a random day, find the probability that she received 3 messages during that time. (d) During a 5-day week, find the probability that there are exactly 3 days when Amy receives no messages between 7:45 and 8:00. [8] 33. The times taken for buses travelling between two towns are normally distributed with a mean of 35 minutes and standard deviation of 7 minutes. Find the probability that a randomly chosen bus completes the journey in less than 40 minutes. 90 of buses complete the journey in less than t minutes. Find the value of t. A random sample of 0 buses had their travel time between the two towns recorded. Find the probability that exactly 6 of these buses complete the journey in less than 40 minutes. [] 34. The number of bus accidents that occur in a given period of time has a Poisson distribution with a mean of 0.6 accidents per day. Find the probability that at least two accidents occur on a randomly chosen day. Find the most likely number of accidents occurring on a randomly chosen day. Justify your answer. Find the probability that no accidents occur during a randomly chosen seven-day week. [] 35 On a particular road, serious accidents occur at an average rate of two per week and can be modelled using a Poisson distribution. (i) What is the probability of at least eight serious accidents occurring during a particular four-week period? (ii) Assume that a year consists of thirteen periods of four weeks. Find the probability that in a particular year, there are more than nine four-week periods in which at least eight serious accidents occur. (0) Given that the probability of at least one serious accident occurring in a period of n weeks is greater than 0.99, find the least possible value of n, n +. [8] 36. The probability distribution of a discrete random variable X is defined by Find the value of c. P(X = x) = cx(5 x), x =, 2, 3, 4. 7

8 Find E(X). 37. The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at any given time are independent of one another and can be modelled using a Poisson distribution with mean 0.2 per day. Determine the probability that there will be exactly four breakdowns during the month of June (June has 30 days). Determine the probability that there are more than 3 breakdowns during the month of June. Determine the probability that there are no breakdowns during the first five days of June. (d) Find the probability that the first breakdown in June occurs on June 3 rd. (e) It costs 850 Euros to service the lifts when they have breakdowns. Find the expected cost of servicing lifts for the month of June. (f) Determine the probability that there will be no breakdowns in exactly 4 out of the first 5 days in June. [3] 38. Over a one month period, Ava and Sven play a total of n games of tennis. The probability that Ava wins any game is 0.4. The result of each game played is independent of any other game played. Let X denote the number of games won by Ava over a one month period. Find an expression for P(X = 2) in terms of n. If the probability that Ava wins two games is 0.2 correct to three decimal places, find the value of n. 39. The distance travelled by students to attend Gauss College is modelled by a normal distribution with mean 6 km and standard deviation.5 km. (i) Find the probability that the distance travelled to Gauss College by a randomly selected student is between 4.8 km and 7.5 km. (ii) 5 of students travel less than d km to attend Gauss College. Find the value of d. At Euler College, the distance travelled by students to attend their school is modelled by a normal distribution with mean km and standard deviation km. If 0 of students travel more than 8 km and 5 of students travel less than 2 km, find the value of and of. The number of telephone calls, T, received by Euler College each minute can be modelled by a Poisson distribution with a mean of 3.5. (i) Find the probability that at least three telephone calls are received by Euler College in each of two successive one-minute intervals. (ii) Find the probability that Euler College receives 5 telephone calls during a 8

9 randomly selected five-minute interval. [2 40. The independent random variables X and Y have Poisson distributions and Z = X +Y. The means of X and Y are and respectively. By using the identity P n Z n PX kpy n k k 0 show that Z has a Poisson distribution with mean ( +). Given that U, U 2, U 3, are independent Poisson random variables each having mean m, use mathematical induction together with the result in to show that Poisson distribution with mean nm. n U r r has a (6) [2] 4. The lengths of a particular species of lizard are normally distributed with a mean length of 50 cm and a standard deviation of 4 cm. A lizard is chosen at random. Find the probability that its length is greater than 45 cm. Given that its length is greater than 45 cm, find the probability that its length is greater than 55 cm. 42. The discrete random variable X has the following probability distribution. P(X = x) = k, x, 2, 3, 4 x 0, otherwise Calculate the value of the constant k; E(X). 43. In a game a player pays an entrance fee of $n. He then selects one number from, 2, 3, 4, 5, 6 and rolls three standard dice. If his chosen number appears on all three dice he wins four times his entrance fee. If his number appears on exactly two of the dice he wins three times the entrance fee. If his number appears on exactly one die he wins twice the entrance fee. If his number does not appear on any of the dice he wins nothing. Copy and complete the probability table below. 9

10 Profit ($) n n 2n 3n Probability (4) 7 n Show that the player s expected profit is $. 26 (2) What should the entrance fee be so that the player s expected loss per game is 34 cents? (2) (Total 8 marks) 0