CHANCE News 6.04

(21 February 1997 to 4 March 1997)


Prepared by J. Laurie Snell, with help from Bill Peterson, Fuxing Hou, Ma.Katrina Munoz Dy, Kathryn Greer, and Joan Snell, as part of the Chance Course Project supported by the National Science Foundation.

Please send comments and suggestions for articles to jlsnell@dartmouth.edu.

Back issues of Chance News and other materials for teaching a Chance course are available from the Chance web site:


Note: We decided to not have a Part 2 for Chance News 6.03 and to just move on to Chance News 6.04. Please do try to answer our lottery mystery in the last item.

Contents of Part 1

Contents of Part 2: A Lottery Bonus

Comments from readers on Chance News 6.03.

John Paulos wrote us about another interesting web site with interactive programs that can be run on the web. This is Kyle Siegrist's interactive version of the gamblers ruin problem His programs are written in Visual Basic script and uses activeX controls. They run fine on Internet Explorer 3.0 for Windows 95. Microsoft has plans to make activeX available for the Mac and Netscape soon. Kyle has also written a book and software (Interactive Probability, Dubury Press 1996). We will review this book in a later issue. Kyle has an NSF project to develop similar modules for introductory statistics.


Elliott Weinstein gave another criticism of the infamous SAT median question. He commented that to assume that n is positive and even is unnecessary and unnecessary information often puzzles a test-taker. Second and more important is that, when n is odd, the quantity in Box B is still well-defined but the quantity in Box A is not defined at all. Again, puzzling information complicates the task for the test-taker.

It seems to us that it is necessary to assume that n is positive and even. There is no agreement on what a median is when a set of numbers does not have a middle value. Also if a is negative Box B could be complex etc.


Paul Alper suggested that, when we discussed the study on the danger of using a cellular phone and driving, we should have mentioned that Efron co-authored a book with one of the authors of the study Robert J. Tibshirani. Alper writes: "While there is no suggestion of collusion, a reference to an expert's opinion should always include possible sources of bias due to a financial or personal connection." On this point, Efron and Kolata were more careful than we were: The NYTimes article stated that "Dr. Bradley Efron, a professor of statistics at Stanford University, said he saw the paper early because he knows Dr. Tibshirani well."


Abe Ross commented on our first discussion question related to the study suggesting that virtual learning might be more effective than classroom learning. We wrote:

(1) The article doesn't say how many students were in the course. How large would the groups need to be for the 20% difference in averages to be convincing?

Abe writes:

This design (as described) has a basic statistical flaw - the students in the real class are not independent subjects. This violates a statistical assumption. Most conservatively they are an N=1. On the other hand, the students in the virtual class are relatively independent and should be counted that way. I'm not sure what the correct method of analysis would be for this design since you end up with very unequal N in the two conditions.


Do you agree that the assumption of independence is more reasonable for the virtual students than for those in class?


Mike Cox wrote that the "New Scientist" commented on the study of Shutte on virtual learning and provided a link to the original paper. The "New Scientist" itself can be read on the web at their web site. You need to register, but it is free.


An article in Part 1 reported that the Federal Aviation Administration (FAA) was going to put on their web page data of airplane accidents and incidents (events that could cause accidents). They put the first databases on Friday and got over 8500 hits on their site. After things calm down it would be interesting to see how the data is presented and how useful it is.

As usual we appreciate these readers' comments.

A probability problem:
American Mathematical Monthly, Feb.1997
Problems and solutions, problem 10576
Proposed by Donald E. Knuth

Alice and Bill have identical decks of 52 cards. Alice shuffles her deck and deals the cards face up into 26 piles of two cards each. Bill does the same with his deck. If any one of Alice's top cards exactly matches any of Bill's, the matching cards are removed. Play continues until none of the cards on top of Alice's piles matches any of the cards on top of Bill's piles. What is the probability that all 52 pairs of cards will be matched?

We played it twice and won both times.


John Lamperti remarked that if I see any pair for Bill that is the same for Alice but in opposite order I might as well not even try to get rid of all the cards. What is the chance of this happening? What other simple things like this could prevent you getting rid of all the cards.

The following article was suggested by Gregg Paulos a former Chance student at Middlebury.

Vaccine is blamed in 125 polio cases.
USA Today, 31 January 1997, A1
Tim Friend

According to the Centers for Disease Control in Atlanta, nearly all US cases of polio since 1980 were due to vaccinations. Of 133 confirmed cases from 1980-1994, 125 were associated with administration of the oral vaccine. A panel convened last fall concluded that a rate of 7-8 cases a year was unacceptable, and recommended a new regime for immunization.

Since 1980, children have been received three doses of oral vaccine by age 2. The new recommendation calls for two injections by four months of age, followed by two oral doses of weakened virus between the ages of 1 and 6. The article says that "the change in policy should eliminate the risk of vaccine induced polio." However, it will increase the annual cost of immunization by $14.7 million.


(1) Do you believe that the risk will actually be reduced to zero?

(2) Is any risk other than zero "acceptable"?

Visual Explanations, by Edward R. Tufte.
Graphics Press, 1996
P.O. Box 430,
Cheshire, CT 06410
$45 Postpaid

This is the third in a series of books written and published by Tufte on the design of the display of information. In the introduction to this book Tufte tells us how these three books are related.

"The Visual Display of Quantitative Information" is about pictures of numbers, how to depict data and enforce statistical honesty.

"Envisioning Information" is about pictures of nouns (maps and aerial photographs for example, consist of a great many nouns lying on the ground). Envisioning also deals with visual strategies for design: color, layering, and interaction effects.

"Visual Explanations" is about pictures of verbs, the representation of mechanism and motion, of process and dynamics, of causes and effects, of explanation and narrative. Since such displays are often used to reach conclusions and make decisions, there is a special concern with the integrity of the content and the design.

"The Visual Display of Quantitative Information" was in the list of 10 bestsellers in the earth's biggest bookstore: Amazon.com. Tufte's previous books have won prizes for their own design and you will find his new book also a visual delight. From his previous books, we also know that we will find wonderful historical examples of the right way to display information and their modern counterparts.

In Tufte's first book we had the graphic display of Napoleon's defeat by the Russian army and winter in the year 1812. Tufte remarked: "it may well be the best statistical graphic ever drawn." The annual weather map of the New York Times provides a modern example of a very informative graph.

In "Visual Explanations" we saw Louis Bretez's wonderful "Plan de Paris," showing every building of this city and a modern counterpart, Constantine Anderson's three dimensional map of midtown Manhattan.

In this new book, we see how John Snow, with his map of the area surrounding the Broad Street Pump, showed the evidence that the cholera epidemic in England in 1884 was caused by the drinking water. He used this map to persuade the authorities to remove the pump handle, an action that was credited with ending the epidemic. Tufte contrasts this with a modern example of the incorrect display of graphic information. He shows us the charts providing information on the failure in tests of the O-rings on space shuttles. These charts played an important role in allowing the Challenger space shuttle to go despite the cold weather. He shows that, if the information on 0-rings had been properly presented graphically, the danger to the O-rings of cold temperatures would have been evident and could have prevented the 1986 Challenger disaster.

A novel chapter, co-authored by Jamy Ian Swiss, a professional magician, illustrates how magicians disguise information and shows that we can learn from them what not to do. This even includes a few hints for our lectures: magicians never explain exactly what they are going to do -- a lecturer should. Magicians never repeat a trick -- an important idea in a lecture is worth repeating.

By now we are supposed to have learned from Tufte to let the data speak for itself and not to litter the graphics with a lot of junk. Alas, as Tufte observes, many of the web-page designers have not learned this. They clutter their homepage with large buttons, flashing lights and other paraphernalia that disguise what in on their web sites. As usual, Tufte is not content to just make critical remarks -- he gives us good examples and shows what should be done.

A common theme of Tufte's books is the proper use of color. Here again we see what a difference it makes, in conveying information, when colors are subtly integrated into the graphics instead of distracting us by their gaudiness.

It would be an interesting study to see if Tufte's impressive books have "made a difference". For example, in his first book his own study showed that, among all the wonderful graphical ways we have to convey information, only maps, time-series plots and bar and pie charts are typically used in newspapers. One seldom sees graphs that relate two or more variables despite the wide use such graphs in scientific publications. Are things any different 20 years later? Our own impression is that the answer is no for the newspapers, but yes for scientific journals, especially medical journals.

In this latest book Tufte shows us what we should be doing when we want to convey information that assesses change, dynamics, and cause and effect, which Tufte regards to be at the heart of thinking and explanation. He does it in a book that is a joy to look at, to read, to think about it, and even to review.

The following article should have been in last month, but it took us a while to track down the source of the article.

The New York Times, 4 Jan. 1997, 1-27
Alan Truscott

Truscott says: "It has long been an article of faith among duplicate players that computer deals are markedly more distributional (more extreme hands) than "normal" deals generated by humans. They are right about that, but wrong in concluding that there is a flaw in computer dealing. It is, we now know the "normal" deals that are abnormal. This point was also made by Persi Diaconis in connection with his study with Dave Bayer of the number of shuffles needed for good shuffling.

This article refers to a study by two Englishmen, Harry Freeman and Len Salmon, who studied 334 duplicate deals played in a English club, over seven months. They reported their findings in "Bridge Magazine" an English magazine published by Chess & Bridge Ltd. who also publish a "Chess Monthly" (web: http://www.chess.co.uk/, e-mail: chesscentre@easynet.co.uk). They kindly faxed us a copy of the Freeman and Salmon article.

Freeman and Salmon analyzed 1336 hand distributions dealt by hand at a bridge club in Abington. They state their main findings as:

1. A markedly reduced occurrence of voids, singletons, and six-and seven card suits.

2. A significant increase in the probability of the most common suit distributions -- in particular, the 4-4-3-2 hand and a consequent decrease in the occurrence of the most common distributions.

It is easier to see why, in rubber bridge, flat hands are more likely in human shuffling. When the hands are picked up after a game, they have clumps of 4 cards of the same suit. If there was no shuffling these would be distributed one to each player. It requires serious shuffling to undo this effect. In duplicate bridge, at the end of a round, most people just put their hands back unsorted and as played and this causes a similar regularity.

To illustrate the problem, Freeman and Salmon consider a "perfect shuffle" carried out as follows: cut the cards exactly in half, and then put one half in each hand and drop the cards, one at a, alternating between the hands. They point out that, if you start with a new deck of cards, do a perfect shuffle, and then deal four hands, each hand will have only two suits. After four such shuffles you find that one player has a great hand: the Ace, King, and Queen of each suit and a Jack!

Freeman And Salmon estimated that human dealing reduced the chance of a preemptive opening by more than 25 percent. When they examined the way that six cards were distributed in two hands, they found that the probability of a 5-1 or 6-0 break was also reduced by about 25%. However, they did not find any significant difference in the distribution of point count.

Freeman and Salmon are not optimistic that bridge players will be persuaded to shuffle better. For duplicate bridge they suggest it would help a lot if the players were simply instructed to mix their hands up before putting them back in the boards at the end of a round.


The authors comment that a Malcolm Grimstom maintained that the way to shuffle a deck efficiently is to first deal it randomly into 5 piles and then re-integrate it before dealing. What does this actually mean? Would it work?

Note: A recent article in "The Economist" has a nice discussion of perfect shuffles and their applications (How to win at poker and other science lessons, The Economist, 12 Oct. 1996, Science and Technology, p. 87.) You will find an elementary discussion of the shuffling results of Diaconis and Bayer in Chapter 3 of Grinstead and Snell's probability book which is available on the web at the Chance Database under "teaching aids".

Some systematic biases of everyday judgment.
Skeptical Inquirer, March/April 1997, pp. 31-35
Thomas Gilovich

Skeptics have long been unimpressed with the reasoning abilities of normal, average people. Before, this opinion was just based on a few observations. But now, psychologists have designed tests to assess these judgments.

The first test is called the "Compared to What?" problem. People like absolute statistics and accept them as true facts. However, sometimes these statistics need to be compared to a set baseline or simply something else in order to see what they really mean.

An example of this occurred in Discover magazine in 1986. The magazine stated that 90% of airplane-wreck survivors had formed a mental plan of escape before the wreck occurred. So, the magazine also recommended that airline passengers be sure to know where all the exits and emergency exits are located and form an escape route. However, the survey never included how many victims planned escape routes. In fact, it would be impossible to find out if forming an escape route really would increase oneıs chances of survival in a plane wreck.

A statistic stated that 30% of all infertile couples who adopt a child eventually conceive a child or children of their own. However, this does not take into account all the infertile couples who do not adopt a child and still conceive a child eventually. Also, if a patient has cancer which goes into remission after mental imagery, would the cancer have gone into remission without the mental imagery or did it truly have an effect?

Other examples include recognizing things. Many people claim to have "gaydar"‹the ability to pick out gay people from a crowd. Though a person may be correct sometimes, this does not necessarily mean that he has picked out all the gay people who crossed paths with him that day.

Many statistics need a control group or baseline for comparison to discover the true meaning behind the number. According to Gilovich: ³The logic and necessity of control groups...is often lost on a large segment of even the educated population.²

Second, we have the "Seek and Ye Shall Find" problem. When people test a hypothesis or theory, they often look more closely at the results which prove them correct. A test for this problem reads as follows:

Imagine that you serve on a jury of an only-child sole- custody case following a relatively messy divorce. The facts of the case are complicated by ambiguous economic, social, and emotional considerations, and you decide to base your decision on the following few observations. To which parent would you award sole custody of the child?

A: Average income, average health, average working hours, reasonable rapport with the child, relatively stable social life.

B: Above-average income, minor health problems, lots of work-related travel, very close relationship with the child, extremely active social life.

Most respondents chose parent B. However, when the question was reworded to ³which parent would you deny custody of the child,² most responded with B as well. Parent B has several advantages and disadvantages, so, when the first version was asked, people looked for advantages(like the close relationship with the child, above- average income) which they found in parent B. When the second version was asked, the people looked for disadvantages and also found them in parent B(like minor health problems, lots of work- related travel). Thus, the people were out to prove the question asked. As in this situation, many people try to simply prove their hypotheses and only look at the proof-worthy evidence.

The third problem is the "Selective Memory" problem. People tend to remember the events that agree with what they expected to happen. However, there are two main forms of events: one-sided and two-sided. An example of a one-sided event is as follows: people believe that the phone only rings while they are in the shower. So, when the phone rings while the person is in the shower, it becomes a memorable event. If the phone does not ring while the person is in the shower, it is not memorable at all. People are more likely to remember a dream if it comes true or applies to their lives. An example of a two-sided event is in a bet from a sporting event: if the team wins or loses, one will win or lose money -- either way, it will be a memorable event.

A test was formed to see if one-sided events truly are remembered more if they coincide with what was expected. A girlıs "diary" was read by several college students. Half was a detailed description of all her dreams and half a detailed description of what happened every day. Half of her dreams were somehow fulfilled while half were not. the students remembered more of the fulfilled dreams than the unfulfilled.

This brings up an important point. Most psychic predictions, etc. are very ambiguous. This is called "temporally unfocused." Thus, they are more likely only to be remembered if they somehow come true. The unfulfilled ambiguous prophecies are easily forgotten. More detailed and exact prophecies, etc.(called ³temporally focused²) are simply more easy to remember whether they come true or not. Another test was formed to evaluate this hypothesis.

Two groups read two different diaries set up much like the one mentioned before, except with prophesies instead of dreams: half prophesies, half real life. However, one diary had temporally focused prophesies which stated exact dates and times of predicted events while the other was more general. Again, half of each groupıs predictions were true and fulfilled, while the other half were not. Both groups remembered about the same amount of fulfilled prophesies. However, the temporally focused diary-group remembered more unfulfilled prophesies than did the temporally unfocused diary-group.

So, these tests showed why most psychic predictions and prophesies are ambiguous and temporally unfocused: if they are temporally unfocused, the listener is more likely to remember only the prophesies which turned out to be true and correct and will probably forget more of the unfulfilled prophesies.

Many other tests exist to determine how poor the average personıs judgment is. However, even through just these few tests, one can see that the average personıs judgment has many shortcomings.


(1) Do these tests really give an accurate view of peopleıs judgments?

(2) Can you give, from your own experiences, an example of a judgment bias of the type described in this article?

A lack of volunteers thwarts research on prostate cancer.
The New York Times, February 12, 1997 A18
Gina Kolata

What is the best way to treat Prostate cancer? As of now, two main options are open to diagnosed men: Radical surgery and careful monitoring. The surgery often leaves men impotent and incontinent, though it also often removes the cancer completely. The monitoring just watches out for accelerated growth of the cancer.

A clinical trial was set up three years ago by the Department of Veterans Affairs in collaboration with the National Cancer Institute to find out which method was most beneficial. The trial included a random designation of which method would be used for the hoped-for 1,050 participants by 1997.

Unfortunately, according to Dr. Timothy J. Wilt, the director of the study at the Minneapolis Veterans Affairs Medical Center, only 315 men have enrolled. Dr. Richard Kaplan, the director of the study at the National Cancer Institute, also mentioned that some of the centers that were going to participate dropped out of the study because of a lack of men willing to participate.

However, this lack of participation does not surprise Dr. Joseph Oesterling, chief of urology at the University of Minnesota. "The reason is that men do not want to let someone else decide on whether they have surgery." He asked 300 men if they would like to enter the study and only four agreed to it. Oesterling says he cannot blame them for feeling this way. "After all, these men are facing a deadly disease... few of them want to have their treatment determined by a coin flip."

However, even Dr. Oesterling agrees that the study is important. According to Kolataıs article, approximately 317,000 men are estimated to be diagnosed with the disease this year and 48,000 will die from it.

The surgery -‹ a radical prostatectomy -- removes the prostate gland which supplies the fluids that nourish sperm. Although the surgery leaves many men impotent and incontinent, supporters argue that removing the infected glad saves menıs lives. On the other hand, skeptics state that prostate cancer usually occurs late in life. According to the article, 95 per cent of all prostate cancer patients are older than 55 and the average patient is in his 60ıs. Also, the cancer usually grows fairly slowly. Thus, there is the possibility that the surgery could be unnecessary and a "monitoring" approach could be preferable. However, the cancer sometimes does grow quickly, and the surgery and even detection could come too late.

Dr. Wilt states that "there is no other way (except the study) to know if surgery prolongs life." He also adds that "selecting treatment is really a best guess" at this time.

Oesterling believes that the study may be coming too late, for people have already decided about their preferred treatment, though the evidence is "less than perfect."

A similar study was conducted in the 1970ıs. However, there was no difference in the death rates of the 61 who underwent the surgery and the 50 who did not. But, Dr. Kramer states that "there really werenıt enough people to test the hypothesis that surgery can save lives."

The most recent study involves men in Sweden. Dr. Jan-Erik Johansson, chief of urology at the Orebro Medical Center in Orebro, Sweden leads this investigation. 223 men deferred treatment and 77 had surgery when the cancer was detected. After 15 years, the study reports a survival rate of 81% for both groups. Johansson believes that "you overtreat many patients in the United States... That is the conclusion of our result."

However, Dr. Patrick Walsh, a urologist at Johns Hopkins University School of Medicine wrote in an editorial accompanying Johanssonıs study that the patients in the Swedish study were different. They were older, and "more likely to die with, and not of, their cancers." Walsh and another doctor, Dr. William Catalona, are firm believers in the idea that surgery can cure early-stage prostate cancer, though neither are participating in the study. Though he notes that having the data from the study would be nice, Dr. Catalona says that "everyone whoıs out there on the front lines really knows it works... If you looked them in the eye and said 'Tell me, are we going to have reliable data on the efficacy of treatment in 5, 10, or 15 years', I think if they were honest theyıd have to say no."


(1) Why do you think it is so hard to get patients to take part in a clinical study when there seems to be genuine uncertainty about the best procedure?

(2) Your Uncle Joe is 72 years old and has been diagnosed with Prostate cancer. He is trying to decide between "watchful waiting" and having an operation. He asked you to help him get the information to make an informed decision. What kind of information would you try to get?


Reader Fred Hoppe was featured in a news article relating his attempt to persuade his students not to waste too much money on lotteries. Before discussing the article it is useful to have some knowledge of the rules of the Canadian Lotto 6/49 in Ontario. Here is how it is played:

A player chooses 6 distinct numbers from 1 to 49. The lottery officials pick 6 such numbers, called "winning numbers" and a 7th different number, called the "bonus number." The bonus number is used only in one situation: if you can match the 6 winning numbers by using the bonus number, then you win the second prize.

Forty-five percent of sales from each Lotto 6/49 is reserved as a prize pool. The total amount of $10 prizes (fifth prize) is deducted from the this prize fund. The remainder is divided into four prize pools. Here are the prizes and your chance of winning them:

Match Win Prob win

all 6 winning nos.(jackpot) 50% of prize pool 1/13,983,816

5 winning nos.+ bonus 15% of prize pool 1/2,330,636

5 winning nos. 12% of prize pool 1/5,5491.3

4 winning nos. 23% of prize pool 1/1,032.4

3 winning nos. $10 fixed prize 1/56.6559

These probabilities and other lotto information can be found on Hoppe's Lotto web site. A good source for additional information about this lottery is http://bud.ica.net/.


(1) Did we get these probabilities of winning right?

(2) You find on Bud's web site the following results for the Feb.26 Lotto 6/49.

Match No. winners Payoff for each

all 6 winning nos.(jackpot) 2 $1,149,548

5 winning nos.+ bonus 9 $76.636.50

5 winning nos. 229 $2,409/50

4 winning nos. 13,933 $75.90

3 winning nos. 277,974 $10

How many tickets were sold? How much did the Ontario Lottery Corporation make on this day? Calculate the expected number of winners of each time assuming all buyers had their numbers chosen by the computer. Are the results consistent with this assumption? What was the expected payoff for a $1 ticket in this lottery.

Professor dashes dreams with anti-lottery course.
The Hamilton Spectator, 5 Feb. 1997, Money section
Lisa Marr

Fred Hoppe, whose statistics class is the largest class at McMaster (900 students), says that the lottery scenario is a great teaching tool. He tries to convince the students that they are wasting their money playing the lottery.

Hoppe says that you are more likely to be struck by lightning than to strike it rich from the lottery. He asks: "Would you pay $1 to bet on 24 heads in a row?" Based on an average $2.2 million jackpot, Hoppe estimates that someone spending $25 a week for 20 years can expect to lose $13,000, about half the $26,000 it would cost to play. Based on a typical jackpot of $2.2 million and assuming there is no sharing of the prizes, in those 20 years a player can expect to win a $10 fifth prize 459 times and a $73.50 fourth prize 25 times. It would take about 2000 years to get a second prize of $131,934 and 40 years to get a third prize of $2,300.

In contrast to this, certified financial planner Robert Beres points out that, if you took just $15 a week and invested it in a mutual fund that earned a 12 per cent average annual return, you'd have $61,735 after 20 years.

While Hoppe thinks some students who buy lottery tickets before taking his course will not after the course, others will continue even after they are better informed about their chances. He reports that not even his mother-in-law takes his advice -- she continues to buy lottery tickets. Hoppe remarks "We're not rational in many ways, that's what makes us human."


(1) On the other hand, doesn't it seem impressive that someone in Canada about once a week, in effect, gets 24 heads in a row tossing a coin? Which is more likely: winning a Jackpot in Lotto 6/49 or getting 24 heads in a row?

(2) What does the probability of getting struck by lightning mean? How is it estimated? (The book "What the Odds are?" by Les Krantz published by Harperperennial is our favorite source for the meaning of such odds and where they come from.)

(3) Can you see how Hoppe gets his 20-year estimates?

(4) How would you try to persuade your Aunt Hattie to quit buying lottery tickets?

Joan Garfield suggested another lottery article. The lottery of interest in this article is a multi-state lottery in the US called Daily Millions. Again it is helpful to know something how the lottery in question works.

The Daily Millions lottery is run by the Multi-State Lottery Association that also runs the Powerball and Tri-West Lotto lotteries. Here is their description of the Daily Millions.

Every night we draw six balls out of three different drums. One drum contains red balls, the second drum contains white balls and the third contains blue balls. Two balls are drawn from each drum. The balls in each drum range from number 01 to 21. Players win by matching 2, 3, 4, 5, or 6 of the numbers drawn. A match occurs when you have the correct color and number for a given ball. The Grand Prize (won by matching all six balls drawn) is paid in cash. Match 5 pays $5,000; Match 4 pays $100; Match 3 pays $5 and Match 2 pays $2.

The jackpot is $1 million dollars. Unlike other lotteries, winners do not have to share the prize with others having the same winning numbers. The one exception is when there are more than 10 winners for the jackpot. In this case the winners share a $10 million dollar prize.

Match                    Win            Probability of win

6 $1 million 1/9,261,000 5 $5,000 1/81236.8 4 $100 1/12,498 3 $5 1/98.6682 2 $2 1/11.1781

More information about this lottery can be found from the Multi- State Lottery Association web page.

On their lottery page they give the following table for the number of winners and the payouts. We assume this is the total since the lottery started up.

Levels                  Winners                    Payout

Match 6                     1                     $1,000,000 
Match 5                   454                     $2,270,000 
Match 4                18,563                     $1,856,300 
Match 3               357,791                     $1,788,955 
Match 2             3,158,957                     $6,317,914 


Estimate the number of tickets sold in this period. About how much did Multi-State Lottery make on lottery during this period? Are the number of winners in each category consistent with your estimate for the number of tickets sold assuming that all buyers have the computer pick their numbers?

Daily Millions beats odds: no one wins -- 5-month losing streak puzzles even statisticians.
Star Tribune, 7 Feb. 1997, 1B
Pat Doyle

The Daily Millions lottery was started nearly five months ago and at the time of this article, 34 million tickets had been sold without a single $1 million jackpot. It is stated that one would expect 3 or 4 winners by now and the probability of having no winners in this period is put at 1/38.

The Daily Millions lottery is run by Multi-State Lottery Association which also runs the Powerball lottery. The Daily Millions was invented to give a lottery where you are do not have to share the jackpot with other winners. You also are about 6 times more likely to win the Jackpot in the Daily Millions lottery than in the Powerball lottery.

However, the fact that no one has won the jackpot yet has hurt the sales. The Daily Millions has slumped from $3.75 million in the first week of the lottery to $1.23 million in the week ending Feb. 1.

Despite not having to pay out any jackpots, the participating states are required to set aside 11 percent of the ticket sales to be put in a pool for future winners. Charles Strutt, executive director of the Multi-State Lottery Association, said the money piling up in the jackpot pool will come in handy if players beat the odds in a different way, winning more than expected.


(1) Did we get the probability of winning the various prizes right?

(2) How did they arrive at the conclusion that the expected number of winners when 34 million tickets are sold is about 4 and the chance of no winner is about 1/38. Does it make any difference that the players do not choose their numbers at random?

P.S. On Saturday February 8 Daily Millions had its first winner.

A lottery mystery.
Chance News 6.04
Laurie Snell

We often hear that people do not actually pick random numbers when they buy lottery tickets. It is claimed that they tend to put down birthdays so the lower numbers are more apt to occur. We decided to check this. We looked at the numbers that were chosen in the Powerball Lottery May 2 and May 3 1996 in some unknown state.

In the Powerball Lottery players choose five distinct numbers between 1 and 45 and then independently choose a 6th number called the "bonus number" again between 1 and 45. Note that this bonus number can be the same as one of the first five numbers. Players have a choice of having the computer choose their numbers or choosing the numbers themselves. If they choose the numbers they first mark 5 numbers in the rectangular array

 1  2  3  4  5  6  7  8  9 
10 11 12 13 14 15 16 17 18 
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 45 36 
37 38 39 40 41 42 43 44 45

They then mark one number for the bonus number in a second such rectangular array. The slip on which they give their numbers actually has ten such rectangles allowing a player to choose up to 5 sets of numbers. On May 2 there were 7,985 sets of numbers chosen by hand and 28,478 picked by the computer. On May 3 there were 17001 picked by hand and 56,496 by the computer.

The first thing we looked at was the distribution of the individual numbers from 1 to 45. The distributions for the numbers chosen by the players on the two different days were remarkably similar. It was easy to reject the hypothesis that they represent samples from a uniform distribution. The smaller numbers were indeed more likely. On both days the most likely number chosen was 7 (about 3.6%) and the least likely 37 (about 1%). Not surprisingly, in the computer picked numbers was very close to a uniform distribution on the numbers from 1 to 45.

We then looked at the individual sets of five numbers chosen by the players ignoring the bonus numbers. For the computer chosen numbers there were no sets of five numbers that were chosen more than 3 times. For those picked by the players on May 3 there were 43 sets of numbers chosen more than 3 times. The largest number of times a set was chosen was 24. This occurred for the set 2 14 18 21 39. The next most popular set was 8 12 24 25 27 chosen 16 times and then 3,13,23,33,43 chosen 13 times. It was not surprising that this last set was chosen so many times since it is an arithmetic progression that could be chosen by just going down a diagonal in the rectangle of possible numbers starting with 3. There were several other arithmetic sequences in our list of popular sets of numbers. The most common arithmetic progressions were those whose differences were 9. They could be obtained by just going down a column in the rectangle.

The mystery was: where did the sets of numbers that were chosen that did not have a recognizable pattern come from? This mystery was pretty much solved when we looked at our grand list of numbers for May 3. We found, for example, that the records from 1771 to 1775 on our list all had the same first five numbers, our most popular set, but different sixth numbers (bonus number). In other words, probably one person decided to fill out the whole slip to buy five sets of numbers differing only in their bonus number. A silly thing to do unless you are concerned only with winning the jackpot! We found the same most popular sets in records 1780 to 1784 suggesting that the same person had done this again. In fact it appears from looking at the records that the same person bought all 24 tickets with the same first five numbers, 2 14 18 21 39 but each with a different bonus number.

This same type of behavior seemed to explain most of the other situations where we got a large number of repetitions of the first five numbers that were not arithmetic progressions. However it left one mystery set: 1 9 23 37 45 which appeared a widely separated points on our grand list 8 times and so, evidently, chosen by 8 different people. So the last remaining mystery is: why this set of numbers? The same set was chosen 4 times on May 2 (smaller set of numbers) and never by the computer.


(1) SOLVE OUR MYSTERY! Why did 8 different people choose their set of five numbers the same set 1 9 23 37 45?

(2) To win the Jackpot in the Powerball lottery you have to your initial five numbers correct plus the bonus number. What is the chance that you win the Jackpot in the Powerball lottery?

(3) In the land of Org there are choose(46,5)*49 = 54,979,155 possible birthdays. How many people do you need in the land of Org to make it a fair bet that two people have the same birthday? (Use your computer or the approximation 1.2*ln(c) where c is the number of possible birthdays (see, for example, P. Diaconis and F. Mosteller, Methods for studying coincidences. J. Am. Stat. Assoc. 84, 853-861 (1989)). What does this tell you about picking lottery tickets for the Powerball Lottery?

Please send comments and suggestions to jlsnell@dartmouth.edu.


CHANCE News 6.04

(21 February 1997 to 4 March 1997)