The first three sections involve isolated events, proportions, and arrivals. The examples from the Lesson are repeated as the first items in the respective sections. A fourth section gives practice in the use of the Poisson Distribution as an approximation to the Binomial Distribution. We conclude with a classic Poisson Problem from real life.
If your answer is a probability, put it in decimal form before clicking on the bar.
1. It has been observed that the average number of traffic accidents requiring medical assistance on the Hollywood Freeway between 7 and 8 AM on Wednesday mornings is 1. What, then, is the chance that there will be a need for exactly 2 ambulances on the Freeway, during that time slot on any given Wednesday morning? The hospital dispatcher needs to know.
2. It is tent inspection time on the Provo Proving Ground, and any irregularities will be cause for extra KP duty. The tent manufacturer's established track record is 1 weaving fault per 50 square feet of canvas. If Barney's tent contains 75 square feet of canvas, and he is otherwise an immaculate tentkeeper, what are his chances of pulling extra KP?
2a. Out of ten such tents from the same manufacturer, how many will have zero faults?
3. Here are the classic 1910 observations of Rutherford, Geiger, and Bateman for the number of alpha particles emitted by a film of polonium, as observed over intervals of one-eighth of a minute (7.5 seconds). The lefthand column gives the number of particles observed in one such time unit, and the righthand column gives the number of units in which such an observation was made:
Number Observed in One Unit Number of Such Units 0 57 1 203 2 383 3 525 4 532 5 408 6 273 7 139 8 45 9 27 10 10 11 4 12 0 13 1 14 1 15 or more 0
From these figures, calculate the rate r, and for that rate, using the Poisson formula, calculate the Expected (E) number of instances of each value. Then compare Rutherford's Actual (A) values, and tell us what you think about the nature of the process behind Rutherford's data.
4. Asa has become everybody's favorite waiter at the Faculty Club. But during his 200-hour probation period he dropped a total of 40 dishes, for a rate of 0.2 dishes per hour. The Chancellor is dining with bigwigs at the Club, and has asked for Asa. They will finish in an hour. What is the chance that their discussion will be jarred by Asa dropping a dish?
4a. Do we know, from the figures alone, that Asa does not usually drop whole trays of dishes?
5. Coliform bacteria are randomly distributed in a certain Arizona river at an average concentration of 1 per 20cc of water. If we draw from the river a test tube containing 10cc of water, what is the chance that the sample contains exactly 2 coliform bacteria?
5a. What is the chance of finding at least 1 coliform bacterium in a 10cc sample of river water?
5b. Do you have a comment on the validity of Problem 5?
5c. Turning from this problem to theory for a moment: Rates are dynamic; they are free to vary in particular cases (and in particular samples). Can you imagine a static equivalent of the 1 in 20 rate r used in Problem 5?
6. According to one source, College suicide rates for 2003 were 1 per ten thousand students. Suppose that Reno College, with 25,000 students, has 4 suicides in 2003. Will the Dean investigate the possibility of a local, nonrandom factor?
7. A batch of cookie dough will be sliced up into 100 cookies and then baked. 400 raisins have been included in the batch of dough, and the dough has been thoroughly mixed so as to randomize the ingredients. What is the chance that, despite these precautions, one or more cookies in the batch will contain no raisins at all?
7a. How many raisins should be put in the batch of dough to be 99% sure that no cookie comes out with no raisins in it?
7b. Comment on the preceding solution from the business point of view.
8. Sam's Party Supplies has had trouble with balloons. Experience has shown that the ones he buys from his present supplier, Rooty Balloons, are 2% defective. Al, representing Tooty Balloons, promises him 1% defectives if he will switch his order to them. Al gives Sam a sample package of 10 Tooty balloons. Sam blows them all up, and one of them turns out to be no good. "Looks to me like we have 10% defectives here," he tells Al. What does Al say in response?
8a. Sam tries nine more packages of ten balloons each, and all of them are free of defects, except one package which has 1 bad balloon in it. Does Sam sign the Tooty contract?
9. Pete's Peppy Pizza Parlor has dismissed its human help, and leased a pepperoni machine which will randomly distribute an average of 20 slices of pepperoni over each of his School Special Pizzas. A busload of 20 kids has just ordered 1 pizza each. How many slices of pepperoni are probably going to be on the most sparsely peppered pizza?
9a. How many are going to be on the least sparsely peppered pizza? And who cares?
10. The switchboard in a Denver law office gets an average of 2.5 incoming phonecalls during the noon hour on Thursdays. Experience shows that the existing lunch hour staff can handle up to 5 calls in an hour. They seem to be well covered. But just to test the edges, what is the actual chance that 6 calls will be received during the lunch period, on some particular Thursday?
11. Security cameras in the Bullion Bank of Nevada have established that, during the last half-hour of business on Friday, an average of 3 customers enter its lobby to deposit their bulk gold. Gold is heavy, and the bank does not want to keep its gold-bearing customers waiting. How many tellers should it have on duty so that the chance that a customer will be kept waiting for a teller is less than 10%?
12. The average number of bad checks received at Exchange Bank of Mariposa is 6 per day. Every morning, upbeat manager Zena bets herself that there will be only 4 bad checks on that day. If she keeps this up for a month, how many times in that month will she probably win her bet?
13. Oahu Airlines has found that 3% of passengers holding tickets fail to show up for departure. It responds by overbooking its flights, all of which use the spacious and maintenance-efficient DC-95, with its 100-passenger capacity. How many extra tickets can Oahu sell without routinely irritating passengers who after all do show up at the terminal?
14. Idaho's Hidden Lake is continually stocked with trout, to a point where guest anglers average 2 trout per hour of fishing. Ardent suitor Sam brings avid trouter Muriel to Hidden Lake Lodge, promising her a rewarding afternoon of sport. Muriel has little patience with incompetence, and will return the ring if in 2 hours she catches nothing. What are the odds that the marriage will take place?
15. Filomena the entrepreneur packages frog legs in small jars for upscale diners, at $100 a jar. She sells them thorough the local supermarket, Filbert's Finer Feedables, where very few people have a taste for exotic cuisine, but on the other hand, very many people turn up in the course of a week. Filbert the manager normally sells 5 jars of frog legs a week. He now says that he will accept no more than 5 new jars each Monday. Filomena argues that he does not want to send even a few of his discerning customers away disappointed, so he should stock enough jars to provide against this possibility. They agree that Filbert will buy just enough jars that there is only a 10% chance that he will sell them all before Saturday night. They cannot figure out how many jars this should be. They turn to you to arbitrate. How many jars should Filbert buy each week from Filomena?
15a. OK, we did the problem and we got the right answer. We are through. We will pass the course, get our degree, and have a career. We can relax. And now that we are relaxing, think a little more about the solution to Problem 15, and comment if desired.
Poisson Approximations to Binomial
16. We are at the county fair. Rick's crooked quarter comes up Heads 80% of the time. He bets Jimmie that he will get 7 or more Heads in the next 10 tosses. Jimmie bets the family horse that Rick will get 6 or fewer Heads. What is Jimmie's chance of riding home from the wager?
16a. What is Jimmie's chance of riding home if the problem is solved in Binomial terms rather than by Poisson approximation?
17. Malingerers Life currently insures 5,000 men aged 42. The actuarial probability that a man aged 42 will die within the next year is 0.0030. It would then seem that the likeliest prospect is that 15 of the insured men will die in the coming year. But there are likely to be variations in any given year. What is the probability that Malingerers Life will have to pay exactly 15 claims on those 5,000 policies during the coming year?
17a. That it will have to pay exactly 20 claims on those policies?
18. Anaconda Federal has taken over Cow Creek Savings, and with it a portfolio of 10,000 mortgages, whose default rate in the past has averaged 6 per month. The Cow Creek portfolio manager, Tom Jones, has been kept on as an interim employee after the merger, and will be given a regular job if he performs well during the transition year. Tom been asked about the prospect for these mortgages. He assures his new boss that, barring some catastrophe, the default rate will not exceed 9 per month. Is it likely that Tom will still have a job 12 months from now?
19. Out of 4,000 invoices paid this year by Bellingham Plumbing Supply, Janice the bookkeeper kited 28 and paid herself the difference in cash. She has been doing this for years, and she has been prudent. She still drives to work in her old jalopy, she still wears her same old sweater, and nobody in this hick town can tell that the Goya over her mantelpiece is real. She feels secure. But on the last day of the year, an auditor walks in, proposing to examine 150 invoices at random. If he audits even one of the kited invoices, Janice will go to jail. The auditor calls for the first random invoice, and Janice has certain options. She can stay at her desk and smile nonchalantly, or she can take off in her jalopy for the Canadian border. She quietly calculates the chance of being discovered, and makes her choice. What does she do?
19a. What would Janice probably have done if she had known this audit was coming?
A Classic Problem
20. During the Blitz, a chart was kept showing V-Bomb hits on South London. The chart was divided into 576 squares, each representing an area 0.25km on a side. Of those 576 squares, the number with a given number of V-Bomb hits was: 1 square with 5 or more hits, 7 squares with 4 hits, 35 squares with 3 hits, 93 squares with 2 hits, and 211 squares with 1 hit, leaving 229 squares without any hits. For most squares, then, the number of hits equals zero, which suggests that this problem may be eligible for Poisson treatment. Compute the value of r, and generate the ideal Poisson Distribution given n = 576 . That set of numbers becomes our Expected number of occurrences. Compare these Expected values with the Actual ones, and judge whether the V-Bomb hits were random or systematic.
20a. Willi Feller, reporting the V-Bomb data in 1968, felt constrained to remark, "It is interesting to note that most people believed in the tendency of points of impact to cluster. If this were true, there would be a higher frequency of areas with many hits or no hit, and a deficiency in the intermediate classes. [The table] indicates perfect randomness and homogeneity of the area; we have here an instructive illustration of the established fact that to the untrained eye, randomness appears as regularity or tendency to cluster." Kalbfleisch in 1979 approvingly repeated the Feller remark. Please comment.
20b. So where are they launching those things from?
"Depend upon it, sir," said Samuel Johnson, "when a man knows he is to be hanged in a fortnight, it concentrates his mind wonderfully." We don't necessarily recommend hanging as a preparation for business life. But we do recommend concentration of mind. How to get this? One hint lies in the fact that people who are exposed to actual situations may have a better sense of actual situations than the statistical consultants that businesses sometimes hire. The factory foreman, plus a few weeks in night school, may be a better statistician, because he begins by being a more experienced deployer of actual resources, than the college graduate with a math major, whose only summer job was shelving books in the library. Both know the numbers, but only one knows what the numbers are about.
In these problems, we have frequently emphasized that Poisson conditions are often not met. Of the 20 numbered problems, only a handful are really valid Poisson situations. Sometimes one can get away with computing Poisson probabilities anyway, and sometimes one cannot. The limit of applicability for Binomial problems handled by Poisson methods can be more or less defined by numbers, so that risk can be provided for. The nature of real situations cannot be defined by numbers, but that end needs a look also. Please try to remember this, before doing any calculations out there in the real world, on which life, liberty, or ledger balance may depend.
24 Aug 2007 / Comments to The Project / Exit to Resources Page