Five probability problems to help us think better

Probabilities are a curious type of numbers. If properly understood, they can help us see the world around us in an appropriate context. Sometimes these numbers can be counterintuitive and downright confusing and sometimes misleading. This article from The Atlantic presents 5 classic problems in probability that can help us think better in probability and statistics. Some of these problems confounded experts. It pays for anyone to pay attention to these problems.

These 5 problems are the Monty Hall problem, the birthday paradox, Gambler’s ruin, Abraham Wald’s memo and Simpson’s paradox.

One common characteristic to all these problems is that they are in some sense paradoxical. The facts on the surface can lead us down one path to a wrong solution. On the other hand, the correct result can be so counterintuitive that it seems absurd.

Take the Monty Hall problem for example. When it appeared in a column authored by Marilyn vos Savant in Parade Magazine in 1990, it drew a great deal of angry responses from readers, some of whom were holders of PhDs on math and statistics (they said so in their disparaging responses). These experts in math and statistics all claimed that the solution proposed by vos Savant was wrong and she should know better. Some of these angry remarks are repeated here.

These experts were wrong! It turned out that even experts can be confounded by probability numbers too. As a result of the controversy, the Monty Hall problem is a probability problem that is known widely and is covered in most standard introductory texts on probability and statistics. The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brainteaser, a book entirely on the subject of Monty Hall problem, is authored by Jason Rosenhouse (Oxford University Press, 2009).

Refer to the above link for a quick introduction to these 5 problems. Three of the problems have been discussed in several math blogs affiliated with this blog. The following are the links to these blog posts.

The following blog posts discuss other classic problems in probability.

Denied airline boarding is a lottery

Think a purchased airline ticket will entitle you a trip to your destination? You board the plane early or on time, does that mean you will be home for supper with your family? The recent online video of a passenger who was booted off an airplane is eye opening. In United Airlines Flight 3411 a passenger, a doctor named David Dao, was dragged off the plane. The video taken by fellow passengers shows a gruesome scene – forcible removal, loud scream, eye glasses askewed and a bloody face. Dao’s belly was visible while the security officers dragged Dao across the floor. Passengers can be heard saying, “my God, what are you doing? …. Look at what you do to him! … Oh My God!”

How often does this kind of violent removals of passenger happen? No often. In fact, the video became viral almost instantly. United Airlines is suffering in the stock market and in the sphere of public opinion, both domestically and abroad, even as far as China. Dao is ethnic Chinese. Many netizens in China wonder whether Dao was chosen to be ejected because of his ethnicity. There is a talk of boycott of United Airlines in China. We do not know how United Airlines determines the passengers for removal or denial of boarding. We will come to this point shortly.

Here’s another good question. How often are airline passengers removed from airplane or denied boarding airplane? Often enough. In fact, airlines have the legal right to remove passengers from the plane or deny a passenger from boarding the plane for any reason, including to vacate a seat to someone else. In the case of Dao, he was removed to make space for an airline crew member. Though the violent outcome shown in the video is extremely unusual, the mere fact that passengers are booted off the plane to make way for others is not unusual.

The airlines industry minimizes the involuntary removals by offering bribe in the forms of cash and free hotel stays. In United Flight 3411, the passengers were offered such incentives. Then the cash incentive was doubled due to the lack of response. Then eventually four passengers were randomly chosen to be removed. Dao was the only one of the four selected passengers who refused, citing that he had to go back home to treat patients.

Come to think of it, being denied boarding or being removed is like a lottery. The lottery ticket is the airline ticket that you purchased. The payout of the lottery is that you will reach your destination later than the scheduled date/time if at all. There is monetary payout for sure, from a few hundreds dollars to a thousand dollars possibly with hotel accommodation (only if you take the bribe). Such lottery is conducted all the time since it is perfectly legal for an airline to overbook. As a result, some “lucky” passengers will be kicked off the flight. If no one takes the bribe in the form of cash/hotel stay, then they select “winners” at random, another similarity with the usual lottery, though it is not known how random the selection is.

What are the “winning” odds in this lottery? According to Department of Transportation numbers, some 46,000 people were “involuntarily denied boarding” by major airlines in 2015. Out of how many “lottery” tickets sold? According to the statistics from the U.S. Department of Transportation’s Bureau of Transportation Statistics (BTS), U.S. airlines and foreign airlines serving the United States carried an all-time high of 895.5 million systemwide in 2015. The odds are 46,000 to 895.5 million or 1 in 194,674, roughly 1 in 200,000.

The “winning” odds are pretty good in comparison with the usual lottery. In a generic lottery where the winning combination is 6 numbers chosen from 49 numbers, the odds for winning are 1 in 13,983,816, roughly 1 in 14 millions. A better comparison is with a smaller lottery. For example, the odds for winning Fantasy 5 in the California lottery are 1 in 575,757, with the odds almost three times longer than the “denied airline boarding” lottery.

California lottery’s SuperLotto Plus has winning odds of 1 in 41,416,353 (roughly 1 in 41 million). For Mega Million, the odds of winning the jackpot are 1 in 259 million. The United passenger David Dao left the plane with a bloodied face and spent times in the hospital in Chicago. It seems that Dao won the jackpot of the “denied airline boarding” lottery.

The incidence was widely reported in social media and in many online news outlets. Here’s a piece from No one wants to win this lottery. Here’s a piece, also from npr, on how not to get bumped and what to do in the event that you are bumped. Here’s a piece on the tone deaf response from United Airlines on the incidence, an indication that they are losing the PR battle.

The following is another video.

How to cut pizza

The title of the post is pizza. But the real story is actually hamburger patties. Knowing how to cut a pizza, or rather knowing the relative size of a slice of pizza will make the story about beef patties interesting. The cutting of the pizza is shown in the following pictures.

Figure 1
This is a whole pizza. Only one slice. The relative size of a slice is 1.

Figure 2
Cut the pizza in two equal slices. The relative size of a slice is one-half or 1/2.

Figure 3
Cut the pizza in three equal slices. The relative size of a slice is one-third or 1/3.

Figure 4
Cut the pizza in four equal slices. The relative size of a slice is one-fourth or one-quarter or 1/4.

Figure 5
Put the pizza slices together from the smallest to the largest.

From the last picture, it is clear that one-third of a pizza is bigger than one-fourth or one-quarter of a pizza. The same relativity would apply for other things too. One third of a loaf of bread would be more than one-fourth of the same loaf. Sitting at the doctor’s office waiting for one-third of an hour would be a longer wait than sitting there for one-quarter of an hour. One-third of a gold bar would be more valuable than one-quarter of a gold bar.

At this point, I hope you agree that one-third of any thing is more than one-fourth of that same thing. In particular, one-third of a pound of ground beef would be more meat than one-quarter of a pound of beef. A hamburger patty weights one-third of a pound would contain more beef than a patty that weights one-quarter of a pound.

According to this article in New York Times, many people think the opposite, that one-third pound of beef is less meat than one-quarter pound of beef! The article is a long one about math educational reform efforts in the United States. The following are the two paragraphs relevant to our discussion.

    One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

    Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray.

The participants in the focus groups believed that 1/3 is a smaller number than 1/4 because the 3 in 1/3 is smaller than the 4 in 1/4. How about the public at large? The fact that the new one-third pounder of Whataburger (the parent company of Whataburger is A&W) was a commercial flop while the new burger was favored at taste tests makes it plausible that this was indeed arithmetic failings on the part of the American consumers.

There are two take-aways. One is arithmetic. The fraction 1/3 is bigger than the fraction 1/4. Or put it another way, the fraction 1/4 is smaller than 1/3. This is visually demonstrated in the series of pictures above. The fraction 1/n refers to the situation of dividing one unit of a thing into n equal pieces. More dividing means each fractional piece is smaller. You can view 1/n as the division of one thing among n people. The more people in the division, the smaller each share is for one person. Thus when the number n in the denominator 1/n gets larger, the smaller the share each person would get.

In the pizza example, the more people want to take a share of the same pizza, each person would get a smaller piece. With one million dollars shared by 2 people, each person would get half a million dollars. But with one million dollars shared by one million people, each person can only get one dollar! Again, when the denominator gets larger, the fraction become smaller.

Another way to know the relative size of a fraction is from using a calculator. One divided by 4 gives 0.25, while one divided by 3 gives 0.3333. Note that 0.25 is smaller than 0.3.

The other take-away is that this example is a vivid example of what author John Allen Paulos called innumeracy, which is the mathematical equivalence of not knowing how to read. In fact, he authored Innumeracy: Mathematical Illiteracy and its Consequences. This book is a good read for anyone who wants to improve his or her numeracy or for anyone who wants to understand the issue of innumeracy. It is not a cure for innumeracy, but is a good start.

The New York Times article mentioned above is authored by Elizabeth Green. Here’s another NY Times article that discusses the article by Elizabeth Green.

Math is a star in the movie Hidden Figures

Hidden Figures is a 2016 biographical drama based on a book of the same name. The movie celebrates the work of three mathematicians/engineers/computer programmers whose work helped propel America into space and win the space race against the Soviet Union. It garnered positive reviews from critics and has been nominated for numerous awards.

Scientists and mathematicians are not hard to find in a place like NASA, currently as well as in the time period in which the story took place (1950s and 1960s). What set these three individuals apart is that they were female and African Americans. They are Katherine Goble Johnson (played by Taraji P. Henson), Dorothy Vaughan (played by Octavia Spencer) and Mary Jackson (played by Janelle Monáe). Thus this movie touches several dimensions – race, gender, history of the space race, and of course the gripping human stories of these three individuals as they struggled to excel in an endeavor they were not expected to excel in.

Math is not just the backdrop of the story; it is front and center in the movie. Of course, the movie does not delve into the details of the math (if it did, it would not gross $129 million worldwide). But the movie is a story of the triumph of math. The math equations worked out by the central characters helped make the space flights safe and successful. There is another sense that it is a story pf the triumph of math.

According to this piece from The Atlantic, math at one point in time was the province only for those who were white and male. Johnson, Vaughan and Jackson and other human computers (later turned mathematicians and engineers) at NASA were allowed to play a pivotal role in the space program because of their math prowess. In fact, due to the societal stereotypes and racial biases of the time, they would normally not even be hired in the first place.

Katherine Goble Johnson was a child prodigy and for a while her talent was underutilized. Then her moment came when NASA needed someone who had skills in geometry and could apply the skills in the calculation for flight trajectories in the Mercury program. The stake was obviously high. NASA was under tremendous pressure to catch up with the Russian. Wrong or inaccurate calculation could mean loss of life and national disgrace. This tension is best dramatized in a scene in which John Glenn wanted to have the numbers checked by the girl (i.e. Johnson). As Glenn was about to be sent off for his orbital mission in 1962, Glenn insisted on having Johnson run through the equations to make sure the trajectory was safe.

Here is a post that focuses on the mathematical achievements of Johnson.

So math is an equalizer. These mathematicians and engineers had a seat at the table because of math. This is an all around wonderful story. This article from Scientific American gives more information on the mathematical and programming work of Johnson, Vaughan and Jackson. This is a piece from NASA on human computers.

Math is as much of an equalizer now as then. I hope the movie inspires youngsters to pay more attention to math and science. Math makes space travel possible. Anyone can learn and excel in math. A whole new realm of possibilities is awaiting for the next generation to explore and unlock.

Due to the popular movie, the story of Johnson, Vaughan and Jackson is now well known. A lot of results come up from Googling their names and the movie. Here’s one piece from Forbes. Here’s the Wikepedia entry about the movie. Here’s two articles (here and here) from

The Story of Billy Barr

I recently came across an article in The Atlantic that tells of a remarkable man named Billy Barr. His claim to fame is his outsize impact on climate science. Thanks to his effort, scientists now have a deeper understanding of the effect of climate changes on the Rocky Mountains and other similar alpine environments. Yet he is not a scientist. He certainly did not set out to go to the Rocky Mountains to become a scientist. The article in The Atlantic is about his remarkable life story. Here’s a video from the National Geographic about Billy Barr.

In the last couple of decades, small but perceptible changes in the high alpine environments had caught the attention of climate scientists. For example, spring snow seemed to melt a little earlier. The flowers blossomed a little sooner. However, they could not make much sense of the changes without a historical context. Without historical data, scientists would not know whether the recently observed patterns were due to random fluctuations or actually represented a clear break from the past.

Barr had made Gothic Mountain in Colorado his home in 1973. Billy Barr did not move from the East Coast to the Rocky Mountains to become a scientist. He went there to find inner peace. He started data recording in his first year there partly as a way to combat boredom and partly to have a point of reference for future winters. He measured snow levels, animal tracks, and in springs the waking of hibernating animals and first joyful calls of bird returning. He filled one notebook, then another and has been doing so continuously for 44 years!

Luckily for climate science, Billy Barr lives very close to the Rocky Mountain Biological Laboratory (RMBL). He became a volunteer there early on in his time in the mountain as a caretaker and later keeping track of expenses as an accountant. Amazingly the scientists did not know about his data collection habit until late in the 1990s!

Once the scientists at RMBL found out, they realized Billy Barr’s data was a priceless trove of data that could help them make sense of the warming trends in the world. Billy Barr’s data are the basis for dozens of research papers on climate science. His data on the high alpine environment has provided an unexpected glimpse into a prior world that scientists never recorded. His notebooks were filled with data on first and last snow, the snowpack levels in between, and when hibernating animals wake and when the birds return after winters.

What is the problem with earlier snow melt and earlier flower blossom? The answer is that such seemingly small changes result in seasonal imbalance in the alpine environment (also called phonological mismatch). A concrete example is the broad-tailed hummingbird. The hummingbird relies on nectar from the glacier lily. Barr had tracked the hummingbird’s return each spring and the first blossoms of the glacier lily. In the past the return of the hummingbird and the blossom of glacier lily were in sync. The glacier lily now flowers 17 days earlier than 40 years ago. If the same warming trends continue, it is likely the broad-tailed hummingbird will completely miss the nectar of the glacier lily, thus spelling the doom for the bird. What happen to the broad-tailed hummingbird will set off a negative chain reaction for butterflies, bees, hibernating mammals, and the other animals that depend on them.

The example of the broad-tailed hummingbird suggests that the ecosystem in the alpine environments might be rapidly approaching a tipping where a small change in the system can result in a drastic change overall.

The snowpack in the Rocky Mountains quenches the thirst in the cities in the surrounding regions. Forty million people rely on the Colorado River for water. Not surprisingly, Barr’s data helps shape water policy for Southwestern region of the United States. For example, hydrologist Rosemary Carroll used Barr’s snowpack data and other sources to model groundwater flows to the Colorado River.

Billy Barr is a fascinating story. For a fuller story, read the Atlantic article or view the National Geographic video. Or you can Google Billy Barr.

Topologist’s sine curve

What is the topologist’s sine curve? Why is this curve attributed to topologists? If you Google Topologist’s Sine Curve, Evelyn Lamb’s article pops up, which does an excellent job of explaining the intuitive idea behind the topologist’s sine curve and why it is an interesting object to mathematicians (this is the link). This is an article she wrote for Scientific American. I chanced upon the article recently. That reminded me of an article I wrote on the topologist’s sine curve in my topology blog (my article).

I will not try to explain too much here, except to say that the topologist’s sine curve is good example of a connected space that is not path connected. Unlike many descriptions of the topologist’s sine curve, the topologist’s sine curve in my article does not use the function \sin (1/x). My topologist’s sine curve is boxy. There are actually more than one curve. The following are two diagrams from my article.

Figure 3 is the closed topologist’s sine curve (closed because it has the vertical bar at the left). This curve is identical (topologically speaking) to the \sin (1/x) plus the vertical bar at the left. In my opinion, the boxy version is better in some way. This curve is constructed by an iterative process (please see Figure 1 and Figure 2 in my article). As a result, the boxy curve brings out the essential idea more clearly – the curve is connected and yet not path connected. The whole curve is connected. Yet you cannot get from the right edge to the left edge.

Figure 4 is the extended topologist’s sine curve, which has an additional bar at the bottom. The bar at the bottom makes it path connected. You can now go from any point in the curve to the vertical bar at the left side.

Please feel free to read any or all of these articles.

Evelyn Lamb’s article also has links to other pieces that she wrote on other interesting mathematical objects (Mobius strip is one).

A Good Will Hunting Story from China


There is an surprising and interesting mathematical story from China. At the center of the story is a migrant worker in China named Yu Jianchun who is generating excitement and awe because of his work in complex and esoteric math problems despite having received no university education, much less formal mathematical training. Facts about the story are still scanty. But he is attracting a lot of attention from academics and from the general public in China and internationally. Here is an article from CNN about Yu. Many people are seeing a parallel between Yu and the character Will Hunting in the Oscar-winning movie “Good Will Hunting”.

After studying at a vocational school, Yu Jianchun became a migrant worker going from place to place working as a parcel delivery man. He always has a passion for mathematics and he spends almost all of his spare time studying it. He is also persistent (and probably stubborn as well). He has spent 8 years working on the problem that currently garners him national and international attention. Whenever he found work in a new city, he always seek out the mathematics professors at the local university in hope of finding confirmation for his math work. He was ignored until a math professor at Zhejiang University, Cai Tianxin, invited Yu in June 2016 to present his math work at a seminar.

The math work of Yu that is generating buzz involves Carmichael numbers, which are odd integers that are prime-like (the usual term is pseudoprimes). For both theoretical and practical reasons, it is critical to test whether a given large odd integer is a prime number. Carmichael numbers are integers that are not prime but yet pass the Fermat’s test for prime numbers. So being able to weed out the Carmichael numbers from the prime numbers will be critical. Testing whether a number is a prime number is more than an intellectual curiosity. Prime numbers are the back bones of encryption systems such as the RSA algorithm, which makes online shopping safe and secure. So the study of prime numbers and Carmichael numbers has implications for information security.

Carmichael numbers are named after R. D. Carmichael who discovered 15 such numbers in 1910. These numbers are very rare. For example, of all the numbers that are less than one billion, there are only 646 Carmichael numbers. In contrast, there are 50,847,534 (over 50 millions) prime numbers below one billion. R. D. Carmichael conjectured that the number of Carmichael numbers is infinite. It was finally proven in 1994 that there are infinitely many such numbers, i.e. there is no upper bound on Carmichael numbers. No matter how big the whole number n, it is proven that there is always a Carmichael number larger than n.

What is special about Yu’s work is that he has discovered a new way to identify Carmichael numbers that is different from the classic algorithm. However, there has not been any precise mathematical statement on what Yu’s results are. So it is hard to get a sense of how special or how significant his results are. In the CNN article referenced above, William Banks, a mathematician who works with Carmichael numbers, seemed to indicate that Yu’s results are about formulas that can generate Carmichael numbers. Yet the same CNN article also indicated that Yu’s work is about an alternative method to verify Carmichael numbers. Is it a new formula for generating Carmichael numbers or is it a new test to check whether a given number is a Carmichael number?

It is likely that Yu’s results are currently being verified. Cai Tianxin, the math professor at Zhejiang University who invited Yu to give a talk, plans to publish Yu’s theory in a book on Carmichael numbers. So in time we will have a better sense of Yu’s achievement.

Another refreshing point about the story of Yu Jianchun is that Carmichael numbers are in the news! Occasionally prime numbers are in the news. For example, when someone finds a new largest prime number or when someone proves a long standing problem about prime numbers. But rarely do we see Carmichael numbers being mentioned by the major news outlets.

It is hard for me to comment on Yu’s work on Carmichael numbers since details are not available. However, Carmichael numbers are an interesting concept in number theory. Here’s an introduction to Carmichael numbers. Here’s an article on how to use Fermat’s test for prime numbers. Here’s another discussion on Carmichael numbers. Finally, here’s another discussion on Fermat’s test.