Topic 8 Binomial & Poisson

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1 1. Spec 91 / S1(new) - Qu 10: Among the blood cells of a certain animal species, the proportion of cells which are Type A is 0.37 and the proportion of cells which are of Type B is Find to 3 d.p. the probability that: (a): in a random sample of 8 blood cells at least 2 will be of Type A (a): in a random sample of 200 blood cells the total number of Type A and Type B cells is at least 81 (b): in a sample of 300 cells there are at least 4 of Type B 2. Jan 92 / S1(new) - Qu 2: Team A has a probability of 2/3 of winning whenever it plays (a): Given A plays 4 games, find the probability that A wins more than half of the games The Coach says they are on a winning streak Let n be the smallest number of games for which there is a 99% chance that one or more of them will have been lost (b): Find n Topic 8 Binomial & Poisson 5. Jun 92 / S1 - Qu 3: The number of accidents per week at a certain intersection has a Poisson distribution with parameter 2.5 Find the probability of: (a): exactly 5 accidents occurring in a week (b): more than 14 accidents occurring in a 4-week period A council officer complains about the intersection, saying that there will be at least a accidents over the 4-week period (c): What is the greatest value of a he can state with 95% certainty 6. Jan 93 / S1 - Qu 2: A shop sells a particular radio at a rate of 4 per week on average. The number sold per week has a Poisson distribution (a): Find the probability they sell at least 2 in a week (b): Find the smallest number that can be in stock to be 99% sure of meeting the demand that week (c): Given the shopkeeper has just sold a radio, find the probability that he doesn't sell another until the same time tomorrow 3. Jan 92 / S1(new) - Qu 3: The number of accidents per week at a factory is a Poisson variable with parameter 2 (a): Find the prob that in any week exactly 1 accident occurs (b): Find n such that they can be 90% sure that there will not be more than n accidents in 1-week The factory is observed for 100 weeks (c): Determine the expected number of weeks in which 5 or more accidents occur (d): Find i such that they can be 90% sure that there will be less than i accidents over the 100 weeks An accident occurs on Monday morning at 9:15 a.m (e): Find the probability that the next accident occurs before 9:15 a.m. the next day (Tuesday) 4. Jan 92 / S1(new) - Qu 9: A manufacturer of Christmas cards produces 1.5 times as many seasonal cards as religious cards Seasonal and Religious cards are randomly distributed in packs of 10 (a): Show probability that a pack has equal numbers of each is (b): Find the probability that the last card packed in the box is the 5th religious card. Why is this less than answer to part(a)? A random sample of 5 packs is taken: (c): Find the probability that exactly three of these contain equal numbers of each type of card A bumper pack contains 50 cards (d): Estimate the probability the pack contains between 24 and 27 (inclusive) religious cards 7. Jan 93 / S1 - Qu 8: A balloon manufacturer produces 40% 'long' and 60% 'round' balloons 5% of all balloons are 'purple' A small pack contains 20 randomly chosen balloons Find the probability: (a): there are equal numbers of 'long' and 'round' balloons (b): more 'long' than 'round' A party pack contains 150 balloons. Find the probability: (c): there are exactly 10 purple balloons (d): there are between 72 and 78 'long' balloons inclusive 8. Jun 93 / S1 - Qu 1: (a): X ~ B(10,0.35). Find p(x 4) (b): Y ~ Po(3.5). Find p(2 < Y 5) 9. Jun 93 / S1 - Qu 8: Small faults occur in the manufacture of a curtain material at an average rate of 0.85 per 10 m² Find probability that: (a): 40 m² of material contains at most 2 faults This curtain material is to be used in 10 rooms Each room requires 40 m² of material Find the probability that: (b): there will be at least 1 fault in the first room's material (c): in half of the rooms the material will contain exactly 3 faults 2% of the hooks on which these curtains hang are defective 500 hooks are purchased (d): Find probability there are between 8 and 12 (inclusive) defective

2 10. Jan 94 / S1 - Qu 6: A chocolate manufacturer produces 3 times as many soft centred as hard centred. Find the probability that, in a box containing 20: (a): An equal number of soft and hard centred (b): Fewer than 5 hard centred A sample of 5 boxes is taken. Find the probability that: (c): exactly 3 of them will contain fewer than 5 hard centred A large box contains 100 chocolates. Find the probability that: (d): it contains fewer than 21 hard centred chocolates 11. Jun 94 / S1 - Qu 4: X is a binomial variable where n = 25 and p = 0.2 Y ~ Po(9). X and Y are independent (a): Find the expected value of 2X-Y (b): Find the standard deviation of 2X-Y 12. Jun 94 / S1 - Qu 8: A random sample of 10 items is to be drawn from a large batch The batch is accepted if the sample contains fewer than 2 defective items and is rejected if the sample contains more than 4 defective items Otherwise, another random sample of 10 items is taken and the batch is accepted if the TOTAL number of defective items is less than 4 (a): Given that the probability of any item being defective is p, show that the probability that the batch is accepted is: q pq p 2 q p 3 q 17 (b): Evaluate this when p = 0.1 As an alternative method a random sample of 75 items is selected from a batch and the batch is accepted if it contains fewer than 10 defective items (c): Find the probability of accepting the batch for the case p = Jan 95 / S1 - Qu 3: An archer fires arrows at a target and for each arrow, independently of all the others, the probability that it hits the bull's eye is 1/8 (a): Given he fires 5 arrows, find the probability that fewer than 2 arrows hit the bull's eye In a competition, the archer fires 5 arrows, collects them from the target and then fires all five again. Find the probability that: (b): on both occasions, fewer than 2 hit the bull's eye (c): his total number of hits in the competition is fewer than 4 If the competition is drawn, then a 'play-off' is conducted The archer fires arrows at a new target with a larger bull's eye (with 25% chance of hitting) until one of them misses the bull s eye. The world record for the 'play-off' is 9 hits in a row and this archer is hoping he can beat this (d): Let n be the smallest number of bull's eye hits before there is a 99.9% chance of 1 or more misses. Find n (e): Comment on whether he has a realistic chance of beating the record 14. Jan 95 / S1 - Qu 7: Telephone calls arrive at a switchboard at a rate of 6 per minute. Find the probability that, in any randomly chosen minute there will be: (a): exactly 2 calls (b): at least 5 calls A call arrives at exactly 10:17 a.m. Determine the probability that: (c): the next call arrives less than 1/2 a minute later (d): Find the probability that in any chosen 10 minute interval the number of calls will be between 55 & 62 inclusive 15. Jun 95 / S1 - Qu 8: Autovend sells new cars and used cars The probability that a sale is a new car is 0.4 5% of all cars sold have automatic gearboxes In a week when 20 cars were sold, find the probability that: (a): at most 5 new cars are sold (b): the last car sold was the 5th new car (c): more new cars are sold than used cars During a 3-month period, 200 cars are sold. Find the probability that: (c): fewer than 85 new cars are sold (d): 12 cars with automatic gearboxes are sold 16. Jan 96 / S1(old) - Qu 9: A painter knows from experience of painting window frames that spots occur on the glass at a rate of 0.4 per thousand square cm The painter paints the frame of a 50 cm by 50 cm window (a): Find the probability that there will be: i} exactly 9 paint spots ii} at least 7 paint spots Four of these frames are painted. Find the probability that: (b): 2 of them will each contain exactly 9 paint spots The painter paints the frame of a 200 cm by 200 cm window: (c): Find the probability there will be at most 175 paint spots 17. Jun 95 / T1(new) - Qu 3: A multiple choice test consists of 20 questions There are 5 choices for each answer James didn t revise for the test so he made guesses for each question (a): Suggest a suitable model for the number of correct answers Find the probability that James got: (b): None correct (c): More than 7 correct A longer exam consists of 50 questions and this time, James revises Assuming that each of his answers has a probability of 0.95 of being correct and given that a prize is awarded for a score of 49 or more (d): find probability James wins a prize

3 18. Jun 95 / T1(new) - Qu 6: A biologist is studying the behaviour of sheep in a large field The field is divided into a number of equally sized squares and the average number of sheep per square is 2.5 The sheep are randomly spread throughout the field (a): Suggest a model for the number of sheep per square Find the probability that a random square contains: (b): No sheep (c): More than 4 sheep (d): Given there are 50 squares in the field, find the probability that more than 5 of them contain no sheep A sheep dog is sent into the field to round up the sheep (e): Explain why the model no longer applies In another field the average number of sheep per square is 20 and the sheep are randomly scattered around the field (f): Find the probability that a randomly selected square contains fewer than 15 sheep 19. Jan 96 / T1(new) - Qu 1: A bag contains a large number of beads of which 45% are yellow A random sample of 20 beads is taken Find the probability the sample contains: (a): Fewer than 12 yellow beads (b): Exactly 12 yellow beads A box contains a large number of discs of which 99.5% are silver (c): Find the probability that a sample of 500 discs contains less than 496 silver discs 20. Jan 96 / T1(new) - Qu 5: Accidents occur at a junction at an average rate of 3 per year (a): Suggest a model for the number of accidents next month (b): Show the probability of 2 or more accidents in the next month is to 4 d.p. (c): What is the greatest number of accidents we can be 95% sure that there will be less than over the next 6 months Residents want a crossing installed The council are to monitor the crossing over 12 months: If there is at least one month with 2 or more accidents in it, they will install a crossing (c): Find the probability that a crossing is installed 21. Jun 96 / T1 - Qu 2: Breakdowns occur on a particular machine at a rate of 2.5 per month Find the probability that: (a): Exactly 3 occur in a particular month (b): More than 10 occur in a 3 month period (c): Exactly 3 occur in each of 2 successive months 22. Jun 96 / T1 - Qu 7: A horticulturalist knows that only 15 in every 100 leaf cuttings will take root (a): In a batch of 10 cuttings, find the probability: i} None take root ii} Fewer than 3 take root (b): Let n be the smallest number of cuttings which need to be examined before there is at least a 95% chance that one or more of them will have taken root i} Show that n satisfies (0.85) n 0.05 ii} Find n (c): Using a suitable approximation, estimate the probability that fewer than six in a batch of 50 cuttings take root 23. Jan 97 / T1 - Qu 4: 5% of bolts produced by a factory are defective. Find the probability: (a): In a box of 20, more than 3 are defective (b): In a box of 20 where exactly 3 were defective, the last 2 packed into the box were defective (c): In a box of 250, between 10 & 14 (inclusive) are defective 24. Jan 97 / T1 - Qu 5: A TV repair company uses a particular spare part at a rate of 4 per week (a): Find the probability that exactly 6 are used in a week (b): Find the probability that at least 10 are used over 2 weeks (c): Find prob that exactly 6 are used in each of 3 consecutive weeks The manager decides to replenish the stock of this part to a constant level n at the start of each week (d): Find the value of n such that the stock will, on average, be insufficient no more than once in a 52 week year 25. Jun 97 / T1 - Qu 7: Frugal bakeries claim pack of 10 of their buns contain on average 75 raisins (a): Suggest a model for the number of raisins in a bun (b): What assumption have you made (c): Show the probability a bun contains more than 8 raisins is (d): Find the probability that, in a pack of 10 buns, at least two buns contain more than 8 raisins (e): Find probability there are more than 80 raisins in a pack of Jan 98 / T1 - Qu 3: A rectangular target is divided into 20 equal squares, seven of which are red Players are blindfolded and given 10 darts to throw at the board Prizes are given for darts that land in red squares Any dart that misses the board is thrown again (a): Suggest a suitable model Find the probability: (b): Fewer than 3 darts land in red squares (c): At least 6 darts land in red squares A skilled darts player asks if he can play without a blindfold (d): What feature of the model in (a) will need refinement

4 27. Jan 98 / T1 - Qu 4: A geography student is studying the distribution of telephone boxes in a large rural area where there is an average of 300 boxes per 500 km² A map of part of the area is divided into 50 squares each of 1 km² and the student wishes to model the number of telephone boxes per square (a): Suggest a model One of the squares is picked at random. Find the probability that: (b): it does not contain any telephone boxes (c): it contains at least 3 telephone boxes The student suggests using this model on another map of a large city and the surrounding villages (d): Comment on the suitability of the model 28. Jun 98 / T1 - Qu 8: Pak-a-Bik manufactures boxes containing 20 biscuits They produce 45% chocolate biscuits and the remainder are plain 5% of biscuits made are wrapped in coloured foil C represents the number of chocolate biscuits in a randomly selected box (a): Give two reasons to support the use of a binomial model (b): Calculate the probability that this box contains: i} Exactly 8 chocolate biscuits ii} More chocolate than plain biscuits The quality manager takes a random sample of 10 boxes (c): Find the probability that exactly 4 of them contain more chocolate biscuits than plain They also produce a large box of 120 biscuits (d): Find the probability this contains i} Exactly 12 biscuits wrapped in coloured foil ii} At least 50 but not more than 60 chocolate biscuits 97.5% of the biscuits contain hydrogenated fats which are bad for one s health e): Find the probability that the large box contains more than 115 biscuits which are bad for one s health 29. Jan 99 / T1 - Qu 5: (a): Give two conditions that must apply when modelling a random variable using the Poisson distribution A kettle is sold by a shop at an average rate of 5 per week The shop manager notices there are 7 kettles in stock at the start of the week (b): Find the probability that the shop runs out of stock that week The manager decides to have enough stock at the beginning of each week to have at least a 99% chance of meeting the demand (c): Find the smallest number of kettles that should be kept in stock at the start of the week (d): Find the probability that the shop sells at least 18 kettles in a 4-week period subject to stock always being available to meet demand 30. Jan 99 / T1 - Qu 7: Articles are produced by a machine and the probability that any one article is acceptable is θ They are packed in small boxes of 5 (a): Show probability a small box contains more acceptable articles than unacceptable ones is: θ³(10-15θ + 6θ²) (b): Find the probability that the last article placed in the box is the second unacceptable article in that box A large box contains 150 articles (c): Assuming θ = 0.85, find the probability that this box contains at least 135 acceptable articles 31. Jun 99 / T1 - Qu 4: It is assumed that each birth in a family is equally likely to be boy/girl (a): Comment on this assumption There are 8 children in a particularly large family (b): Given there are at least 2 boys, find the probability that there are more boys than girls (c): What assumption have you made 32. Jun 99 / T1 - Qu 7: A pottery produces large quantities of drinking mugs, 5% of which are 'seconds'. A sample of 20 is taken (a): Suggest a model for the number of 'seconds' (b): Find the probability that there are: i} Exactly 2 'seconds' ii} More than 4 'seconds' A sample of 150 is taken (c): Find the probability that the number of seconds in this sample is between 12 and 15 inclusive using: i} A Poisson approximation ii} A Normal approximation (d): State why an adjustment had to be made for part (ii) Without any approximation, the answer to part (c) would be to 4 d.p. (e): Comment on your answer to part (c) 33. Jan 00 / T1 - Qu 2: Dandelions occur in a lawn at a average rate of 0.4 per m² (a): Suggest a model for X = no of dandelions in 1 m² of lawn (b): Find p(x > 2) 10 lawns each of area 1m² are investigated. Find probability that (c): at least one of them contain more than 2 dandelions (d): Find probability that 20m² of lawn contains exactly 9 dandelions 34. Jun 00 / T1(old) - Qu 3: The probability that one of the engines in a model aeroplane fails is 0.2, independently of the other engines A flight is successful unless more than half of the engines fail Determine which of a 2-engine model, a 4-engine model or a 6-engine model has a higher probability of making a successful flight

5 35. Jun 00 / T1(old) - Qu 8: A student is known to make an average of 1.5 errors per page (a): Suggest a model for the number of errors per page (b): Find the probability that a page contains more than 4 errors She produces a 4-page essay (c): Find the probability that it contains exactly 10 errors She produces a project containing 30 pages (d): Find the probability it contains more than 50 errors Another student has written a 65-page report He wants to estimate the number of words it contains (e): Suggest two factors to consider when determining the population to sample (f): Give one advantage and one disadvantage associated with using the 'page' as his sampling unit rather than the 'line' 36. Jan 01 / T1(old) - Qu 4: (a): Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution A college student who glanced at a newspaper headline thought it said that 1 in 20 students suffers sleeping problems (b): There are 20 students in his class. Find the probability that more than 4 of them suffer from sleeping problems In fact, it said, "1 in 200 students suffer sleeping problems" (c): Find the probability that, in a group of 10 students, exactly 3 of them suffer sleeping problems There are 1500 students in the college (d): Find the probability that fewer than 7 suffer sleeping problems 37. Mock 00 / S2(new) - Qu 6: On a weekday morning customers arrive at a village post office at an average rate of 3 per 10 minute period. Find the probability that: (a): At least 4 customers arrive in the next 10 minutes (b): No more than 7 arrive between 1:00 a.m. and 11:30 a.m. The period from 11:00 a.m. to 11:30 a.m. next Tuesday morning will be divided into 6 periods each of 5 minutes (c): Find the probability that no customers arrive in at most one of those periods The post office is open for 3.5 hours on Wednesday mornings (d): Estimate the probability that more than 49 customers arrive during that time (e): What assumption has been made throughout this question 38. Jun 01 / S2(new) - Qu 2: On a motorway accidents occur at a rate of 0.9 per month (a): Show the probability of no accidents in the next month is (b): Find the probability of exactly 2 accidents in the next 6 months (c): Find the probability of no accidents in exactly 2 of the next 4 months The council decide to monitor the motorway for 5-months (d): Find the probability that the next accident doesn't occur until 2-months after they start the monitoring (e): What is the least number n that they can expect with 95% certainty that there will be less than n accidents 39. Jun 01/ S2(new) - Qu 4: A company sends 20% of its letters by 1st class post. In a sample of 10 letters, find the probability that the number posted 1st class is: (a): At least 3 (b): Fewer than 2 On Monday there are only 12 first class stamps, but 70 letters to be posted (c): Find the probability that there are enough 1st class stamps (d): State your assumption 40. Jan 02 / S2(new) - Qu 3: An airline knows that 3% of passengers do not turn up for its flights The airline adopts a policy of selling more tickets than there are seats on the flight, so for an aircraft with 196 seats, they sold 200 tickets (a): Suggest a model for X = no of passengers that don't turn up Find the probability that: (b): More than 196 turn up (c): There is at least 1 seat empty The director decides to revise the policy They still sell 200 tickets, but decide to change the number of seats in the aircraft so that the chances of 1 seat being empty is less than 0.5% (d): How many seats should the aircraft now have 41. Jan 02 / S2(new) - Qu 5: An internet provider says, on average, 3 users every hour fail to connect 1st time (a): Give two reasons why Poisson might be a suitable model for the number of failed connections per hour (b): In 1 hour, find the probability that i} All users connect at their 1st attempt ii} More than 4 users fail to connect at their 1 st attempt (c): Write the distribution for the number that fail to connect at the 1st attempt in an 8-hour period (d): Find the probability that 12 or more users fail to connect in an 8-hour period (e): The boss of the firm wants to know the least number of users n that he can claim with 95% certainty that less than n users will fail to connect at their 1st attempt in any 8-hour period 42. Jun 02 / S2(new) - Qu 6: Faults occur in twine at a rate of 1.5 per 25 m (a): Find the prob that a 25 m length contains exactly 4 faults Twine is sold in balls of 100 m. A customer buys 3 balls (b): Find prob only one of them will have fewer than 6 faults A giant ball containing 500 m is produced (c): Find probability it will contain between 23 & 33 faults inclusive A giant ball of twine is examined as it is slowly unrolled (d): Find the probability that the first fault is found after 25 m have been unrolled

6 43. Jan 03 / S2 - Qu 3: The number of weeds growing in a meadow is to be modelled using the Poisson (a): What conditions must apply for the model to be valid Assuming a mean of 0.7 weeds per m², find the probability that: (b): there are fewer than 3 weeds in a 4 m² plot (c): there are more than 66 weeds in a plot of 100 m² 44. Jan 03 / S2 - Qu 5: The probability that an egg has a 'double-yolk' is 0.05 Eggs are packed in boxes of 10. Find the probability that in a box the number of eggs with 'double-yolks' will be: (a): Exactly one (b): More than 3 A customer brought 3 boxes (c): Find the probability that only 2 of the boxes contained exactly 1-egg with a double-yolk The farmer delivered 10 boxes to a local shop (d): Find the probability that the delivery contained at least 9-eggs with double yolks The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g (c): Find the probability that an egg weighs more than 68 g 45. Jun 03 / S2 - Qu 2: (a): Under what conditions can a Poisson distribution be approximated using the Normal distribution (b): Y ~ Po(30). Estimate p(y > 28) 46. Jun 03 / S2 - Qu 3: In a town, 30% of the residents listen to the local radio station Four residents are chosen at random (a): Specify the distribution of X = the number of these 4 that listen to the local radio station (b): On graph paper, draw the p.d.f. of X (c): Write down the most likely number of these 4 residents that listen to the local radio station (d): Find E(X) and Var(X) 47. Jun 03 / S2 - Qu 4: (a): Write down the conditions when the binomial distribution may be a suitable model A six-sided die is biased. The number 5 is twice as likely to appear as any other number. All other faces are equally likely to appear The die is thrown repeatedly. Find the probability that: (b): The first 5 will occur on the 6th throw (c): In the first 8 throws there will be exactly three 5s 48. Jan 04 / S2 - Qu 2: (a): R ~ B(12, 0.35). Find p(r 4) (b): S ~ Po(2.71). Find p(s 1) (c): T ~ N(25, 5²). Find p(t 18) (d): U ~ B(60, 0.85). Find p(u = 10) (e): V ~ Po(40). Find p(v 35) (f): W ~ B(100, 0.15). Find p(w 13) 49. Jan 04 / S2 - Qu 3: Given X ~ B(n, p) (a): Write down the value of p that will give the most accurate estimate when approximating using the normal distribution (b): Give a reason for your answer (c): Given n = 200 and p = 0.48, find p(90 X < 105) 50. Jan 04 / S2 - Qu 4: (a): Write down two conditions needed to be able to approximate the binomial distribution using Poisson A researcher said 1 in 150 people is likely to catch a virus (b): In a sample of 12 people, find the probability that 2 of them catch the virus (c): In a sample of 1200 people, find the probability that fewer than 7 of them catch the virus (d): Comment on the assumption made to apply this model 51. Jun 04 / S2 - Qu 4: (a): Under what conditions can the Binomial distribution be used as a model 10% of electronic components are defective A batch contains 20 components (b): Give a model for the number of defective components in a batch Find the probability that a batch contains: (c): No defectives (d): More than 6 defective (e): Find mean and variance of the no of defective components A supplier buys 100 components. They will refund if there are more than 15 defective (f): Find the probability that a refund will be issued 52. Jun 04 / S2 - Qu 6: Defects occur in a carpet at a rate of 0.05 per m² (a): Suggest a model for the number of defects 30 m² is laid in a hotel foyer. Find the probability this contains: (b): Exactly 2 defects (c): More than 5 defects A total of 355 m² was laid in the whole hotel (d): Find the probability there are 22 or more defects