Chance News 2
June 1 2005 to June 30 2005
Dan Rockmore's book: Stalking the Riemann Hypothesis
Stalking the Riemann Hypothesis
Pantheon Books, New York, 2005
The Proof: an interview with Dan Rockmore
New Hampshire Public April 12. 2005
As the stakes increase, Prime-Number theory Moves Closer to Proof
Wall Street Journal, Science Journal, April 8. 2005
The Telegraph (Calcutta, India), April 8, 2005
In 1998 the Mathematical Sciences Research Institute in Berkeley, California had a three-day conference on "Mathematics and the Media". The purpose of this conference was to bring together science writers and mathematicians to discuss ways to better inform the public about mathematics and new discoveries in mathematics. As part of the conference, they asked Peter Sarnak, from Princeton University, to talk about new results in mathematics that he felt the science writers might like to write about. He chose as his topic "The Riemann Hypothesis" This is generally considered the most famous unsolved problem in mathematics and is the major focus of Sarnak's research.
In his talk, Sarnak described some fascinating new connections between the Riemann Hypothesis, physics and random matrices. He used only mathematics that one would meet in calculus and linear algebra. Sarnak's lecture, and a discussion of his talk by the science writers, can be found here under "Mathematics for the Media"
Unfortunately, Sarnak's talk was not fully appreciated by the science writers. The first writer to comment said that she felt like she did when she was in Germany and a friend took her to a party. At the party the Germans initially tried to speak to her in English, and she could understand them pretty well, but then they would drift into German and she was lost.
Another science writer said that, to write an article based on Sarnac's talk, she would have to sit down with him and have him explain what he had said in a way she could understand. Then she would have to write an article for her readers without using formulas or mathematical symbols--a few graphics would be o.k.
With his book "Stalking the Riemann Hypothesis", Dan discusses the Riemann Hypothesis and its history in a way that the science writers proposed for the general public. He does this by explaining the relevant mathematical concepts in terms of concepts familiar to his readers. For example the rate of increase is discussed first in terms of the spread of a rumor and the logarithm in terms of the Richter scale. Density is described in terms of population density noting that the average number of people per square mile living in South Dakota is quite different from that of New Jersey. Then we read
Similarly, we can ask how many prime numbers "live in the Neighborhoods" of a particular number? Gauss's estimates imply that as we traipse along the number line with basket in hand, picking up primes, we will eventually acquire them at a rate approaching the reciprocal of the logarithm of the position that we've just passed.
While there are no formulas and minimal math symbols, Dan does make good usage of great graphics made by our mutual friend Peter Kostelec.
Along the way, Dan gives a lively discussion of the great mathematicians Euler, Gauss, Riemann and many others up to present working on solving the Riemann Hypothesis. He also gives the readers an understanding of what mathematics and mathematical research is all about. In the Telegraph Dan is quoted as saying:
My overarching goal has always been to provide a window through which the public might get some sense of what mathematical research is like while showing the surprising breadth of math as a subject, I felt that the best way to achieve this was through a single story, for I believe that narrative is the key to holding a reader's interest. It seemed to me that the history of the Riemann Hypothesis - given its central place in modern mathematical history and research - would provide such an opportunity. The story of the search for its resolution would provide a structure off which I could hang a broader story of modern math.
This is not Dan's first attempt to give the general public a better understanding of what mathematicians do, what mathematics is all about and how it effects our daily life. (I once mentioned to a musician that I was a mathematician and he said, "I guess you learn how to multiply and add numbers faster than we can".)
Dan's first article for the general public was an essay in the New York Times in 1998. In this essay Dan used the movie" Good Will Hunting" to illustrate stereotypes of mathematicians. He writes:
The main messages of the movie are old and trite: You are either someone who can do math or you are not; mathematics is impossible to explain to others, even other mathematicians, and to be a mathematician and to think about mathematics is to separate yourself from most of society. After all, what could be more at odds with Will's working-class, street-wise background than a talent for mathematics?
Dan then gives examples from the arts to show the pervasiveness of mathematics in the real world. For example, the Movie "Sliding Doors" illustrates how long-term behavior of a system can be highly dependent on the initial conditions -- the butterfly effect. He remarks that identifying these connections does not need to lead to madness as it does for the hero of the movie Pi.
More recently Dan has presented on Vermont Public Radio a series of mathematical commentaries with jazzy titles such as "Math, a love story", "Halving Your Cake" and "Can you hear the shape of your date?" giving explanations of important modern mathematical results that we see in everyday life. You can hear these commentaries here.
Last year Dan and his colleagues Wendy Conquest and Bob Drake made a movie called "The Math Life" in which leading mathematicians talk about how they got into mathematics, what kind of mathematics interests them and what it is like to work on and solve a mathematical problem. This movie was shown on public television and used in numerous classrooms.
"Stalking the Riemann Hypothesis" is Dan's greatest challenge to bring mathematics to the general public. It is hard to image a better mathematical story to show the general public the beauty, the excitement, and the importance of mathematics. Dan is an engaging writer and he tells the story in a wonderful way. We can hardly wait for the movie!
The powerball lottery suspects fraud, but it's the Fortune Cookies
Who needs Giacomo? Bet on the fortune cookie
New York Times, May 11, 2005, National desk; Pg 1
This article reports unexpected winnings in the Powerball lottery of March 30, 2005 which lottery officials thought might be fraudulent but which had a much simpler explanation.
For the Powerball lottery, a player chooses 5 distinct numbers from 1 to 53, which we will call the "basic numbers." In addition, the player chooses another number between 1 and 42 which we will call the "bonus number." The lottery randomly chooses 5 basic numbers and one bonus number. If your 6 numbers agree with those chosen by the lottery you win the jackpot (a huge amount). If there is more than one jackpot winner, you share the jackpot with the other winners. There are 8 additional prizes which you do not have to share with other winners. For example, if you buy a $1 ticket and your 5 basic numbers match those of the lottery but your bonus number does not, you win $100,000.
When you buy a $1 lottery ticket, for an additional $1, the lottery offers another bet called the "Power Play". For the Power Play, the lottery randomly chooses a number from the numbers 2,3,4,5,5. This number is called the "multiplier". If you make this bet and you win any prize other than the Jackpot, the prize is multiplied by the multiplier.
On March 30 drawing of the Powerball lottery, 110 players made a $1 bet, choosing as their five basic numbers 22,28,32,33,39 and as their bonus number 40. The lottery chose the same five basic numbers but chose 42 for their bonus number. The lottery chose 5 for the multiplier. 89 of the 110 winners did not choose the Power Play and so each won $100,000. 21 players chose the Power Play and, since the multiplier was 5, they each won $500,000 dollars. Thus the lottery paid out 19.4 million dollars to these winners. Actually, they didn't have to pay out this much since on the back of a ticket:, in small print, we read:
In unusual circumstances, the set prize amount may be paid on a pari-mutual basis, which will be lower than the published prize amounts.
Evidently the Lottery officials decided not to use this option in this case. In addition the article states that the Lottery keeps a $25 million reserve for odd situations.
Power ball officials stated that, considering the number of tickets sold in the 29 states, they expected 4 or 5 winners. The article quotes Chuck Strutt, executive director of the Multi-State Lottery Association as saying:
Panic began at 11:30 pm. March 30 when he got a call from a worried staff member. We didn't sleep a lot that night. Is there someone trying to cheat the system?
The lottery athorities tried a number of theories about how people choose their numbers. For example many players pick their numbers following a geometric design on the ticket. Nothing worked. But then the first three winners said that they had obtained the numbers from a fortune cookie. With this lead, they just had to find the fortune cookie maker who had the winning numbers. They found that many different brands of fortune cookies come from the same Long Island City factory owned by Wonton Food. This company turns out four million fortune cookes a day, which are delivered to dealers over the entire country. When shown the numbers, Derrick Wong, of Wonton Food, verified that they had used these numbers. The numbers were chosen from a bowl but the company plans to switch to having them chosen by a computer and Derrick plans to start playing the lottery.
(1) The articles says:
Of course, it could have been worse. The 110 had picked the wrong sixth number -- 40, not 42 -- and would have been first-place winners if they had.
Worse for whom?
(2) How do you calculate the probability of getting the 5 basic numbers but not the bonus number correct?
(3) Do you think that the sales of fortune cookies will increase?
Red enhances human performance in contests
Red enhances human performance in contests
Nature, Vol. 435, May 19, 2005
Russell A. Hill, Robert A. Barton
Hill and Barton examined the results of the 2004 Olympic games in four categories of competition—boxing, tae kwon do, Greco-Roman wrestling, and freestyle wrestling—chosen because in each match, one contestant wears red, the other blue. Within weight classes, color assignments are apparently made randomly in the first official round of competition; subsequent assignments are then determined by the initial roster. (In boxing, for example, the winner of bout 1 plays in red in the second round against the winner of bout 2, who plays in blue. The same arrangement is used for the winners of bouts 3 and 4, 5 and 6, and so on.) The authors found that for each sport, significantly greater than 50% of bout winners wore red outfits.
To study the findings more closely, Hill and Barton focused on competitions in which the two contestants were most evenly matched. According to the article, they did this because "wearing red presumably tips the balance between losing and winning only when other factors are fairly equal." Such matches do appear to represent the only cases in which there were significantly more red than blue winners. (A description of their methods can be found in a supplementary text file at Nature’s website—see below.)
The authors, both members of the Evolutionary Anthropology Research Group at the University of Durham in the UK, evidently favor a behavioral/biological explanation for the apparent red advantage: "Red coloration is a sexually selected, testosterone-dependent signal of male quality in a variety of animals, and in some non-human species a male’s dominance can be experimentally increased by attaching artificial red stimuli. Here we show that a similar effect can influence the outcome of physical contests in humans."
DISCUSSION QUESTIONS :
(1) Assuming that Hill and Barton’s results are valid, what do you think is the most likely explanation for the surplus of red winners?
(2) The authors don’t discuss the possibility that the color of competitors’ outfits affects the performance of the contest judges. (Judges assign points to each athlete during the match.) How might you evaluate this possibility?
(3) Get the data and carry out your own test to see if there is a signficant difference between the reds and the blues. Red enhances human performance in contests
Mix math and medicine and create confusion
Mix math and medicine and create confusion
New York Times, April 26, 2005, F 11
Richard Friedman, M.D
This article provides an interesting exchange between a doctor (Doctor Friedman) and a statistician (Judith Singer). We give the entire exchange as presented in this article by Dr. Friedman.
Patients may not know it, but there are two questions that make doctors cringe. The most common is, If you were me, which treatment option would you pick? The tougher one is, What are the chances that this treatment will help me?' Both questions cut to the heart of medical decision making and involve assessing risk and probability, which does not come naturally to many people.
For example, a depressed patient told me she had read that the chances were 60 percent that she would respond to the antidepressant I had prescribed for her.
"That means that 60 percent of the time I will feel better on this, right?" she asked.
Well, not exactly. I explained that if 10 people with a depression just like hers walked into my office, about 6 would be expected to respond to that antidepressant.
But the statistics, I told her, referred to a large sample, not an individual. She would either improve with this treatment or she would not, I said, but she shouldn't worry because we would keep trying until we found a treatment that worked.
"You mean my chances of getting better are really only 50 percent?" she asked with dismay.
Dr. Judith D. Singer, a statistician and professor at the Graduate School of Education at Harvard, explained "You and your patient are confusing two different concepts. The number of possible outcomes -- in her case either responding or not responding to an antidepressant -- has nothing to do with the actual probability of either outcome happening."
For example, Dr. Singer said, "Either a woman is pregnant or not. She can't be a little pregnant. But that doesn't mean that she has a 50 percent probability of being pregnant."
A woman who takes a fertility pill may stand a much higher chance of actually getting pregnant than if she goes without it. If my patient was typical of the subjects in the clinical trial she read about, Dr. Singer said, "she is more likely than not to get better on that antidepressant."
(1) The doctor explained 60% chance of a response from the antidepressant as "If 10 people with a depression just like hers walked into my office, about 6 would be expected to respond to that antidepressant. The statistician's explanation was: If my patient was typical of the subjects in the clinical trial she read about, she is more likely than not to get the better on that antidepressant. What are the pros and cons of these explanations? How would you answer the question?
(2) If your doctor would answer one of the two questions that make doctors cringe, which would you prefer?
Marilyn answers a lottery question
Parade, April 10, 2005
Marilyn vos Savant.
A reader poses the following question:
My wife and I attended a "reverse raffle," in which everyone bought a number. Numbered balls were then drawn out of a bin one at a time. The last number would be the winner. But when the organizers got down to the last couple of dozen balls, they discovered that some numbered balls had been overlooked. So they added those balls to the bin and continued the drawing. Didn’t the added balls have a much better chance of winning?
Marilyn responds, "Yes, they did. But because everyone had an equal chance of his or her numbered ball being one of those overlooked, the last-minute addition made no difference to anyone’s chance of winning. The raffle was still fair."
Marilyn's answers to probability problems often stir up controversy, and this was no exception. The discussion continues in the column below.
Parade, June 5, 2005
Marilyn vos Savant.
A reader had this objection to Marilyn's answer:
Marilyn: I disagree with your answer. Participants whose balls were left out had a higher likelihood of winning. Regardless of whether they had a fair chance of being overlooked, the raffle was not mathematically fair. Assume there were 20 participants. The odds of winning should be one in 20 throughout the game for each contestant. Put 15 balls in a jar and withdraw 10. Then add the missing five. The first 15 balls had a two-in-three chance of “not winning” until the five balls were added. The missing five balls had a zero chance of “not winning” during that time, then had a one-in-10 chance of winning after they were added. Only the five balls that were in the bowl the entire time had a one-in-20 chance of winning.
Marilyn says her original answer was correct, and asks the reader to "consider a scenario in which the added balls were withheld (on purpose) instead of overlooked." She says. "Your explanation works in that case. So it cannot work in the case when the added balls were merely overlooked."
(1) Do you understand Marilyn's last response?
(2) The reader is actually giving conditional probabilities. Do you agree with their values?
(3) Now consider the reader's set-up, under Marilyn's original assumption that each ball had an equal chance of being overlooked in the first stage. Thus, there are 20 balls, and 15 are initially selected at random and placed in the bin. Now 10 are drawn one at a time at random from the bin. At this point, the 5 balls originally omitted are added to the bin. Then balls are drawn one at a time at random from the bin. Looking at the whole process, does each ball have a one-in-20 chance of being the last ball in the bin?
(4) Upon realizing the correct answer to Marilyn's problem, a chance news reader suggested that this meant that the famous 1970 Vietnam lottery was fair also. Recall that in this lottery 366 balls with the possible dates in a year were put in a bowl and mixed up. Then the balls were drawn out one at a time and the dates on the balls determined the order in which draftees would be called up. It was estimated that those with birthdays were on the last third of the balls drawn would not be called up at all. Å statistical analysis suggested that the balls were not well mixed and as a result those born in the early months were significently more likely to be called up than those born in the later months. But our reader said that, since our birthdays are random, the lottery was still fair. Do you agree?