Tanya Khovanova's Math Blog

It is unfortunate that crooks understand probability. Here is a scam that was very popular back in Russia.

A bad guy pretends that he has a close relative on the hiring committee of a college. He takes bribes from prospective students, promising to help them pass the entrance exams at this college. He doesn’t guarantee the admission, but he guarantees the money back. After getting the money, he does nothing. If the student passes the entrance exams, he keeps the money. If not, he returns the money. Simple probability — someone will pass the exams by chance, making him a lot of money.

Here is another Russian scam. This time the crooks have some understanding of conditional probability. These “psychics” promise to correctly predict the gender of your future child. They tell you a random gender, but for their bookkeeping they file the opposite gender. This way, even if you complain, they still keep your money. They show you their books and pressure you into believing that you misunderstood, misheard or misremembered the answer. The probability that you complain if they are right is zero.

Let us all learn probability theory to recognize scams and not fall for them.

China Girls Math Olympiad is becoming an international math Olympiad for girls. When I first heard about this competition I felt very sad. I need to explain myself here.

For many years I felt very proud that math Olympiads do not separate the genders. Most Olympic sports, like running or swimming, have separate competitions for men and women. I felt that joint competitions for math demonstrated the spirit of equality in our math community. I felt that insofar as gender didn’t matter, mathematics was more democratic than other sports.

At the same time I do understand how people might assume for the following reasons that a math competition among only girls would be useful:

• They promote the idea of math to girls.
• They can help girls who are into math to feel less lonely.
• They generate additional resources and training for girls.
• They might be less stressful for some girls, than mixed math competitions.
• They help promote the image of female mathematicians to society
• They provide further opportunities for girls to earn prizes and improve their resumes.

See also the article: First US Team to Compete in the China Girls Mathematical Olympiad.

On the other hand, this development scares me. If we have a separate girls Olympiad, will that soon lead us to have two Olympiads, one for boys and one for girls? Two separate Olympiads would be a defeat for women mathematicians. Or, maybe I shouldn’t be scared. The percentage of girls at the most prestigious mathematics competition, the International Mathematical Olympiad, is so small that it can be viewed as virtually boys-only.

Mathematics is becoming similar to chess. There is a World Chess Championship where both men and women are allowed to compete, and there is a separate Women’s World Chess Championship. The interesting part is that Judith Polgar, by far the strongest female chess player in history, never competed in the Women’s World Chess Championship. I suspect that I understand Judith. She probably feels that women-only competitions diminish her, or that chess is about chess, not about gender. In any case, I hope that one day the separate girls Olympiad will not be needed.

Browsing Braingle I stumbled upon a standard probability puzzle which is very often misunderstood:

Suppose I flip two coins without letting you see the outcome, and I tell you that at least one of the coins came up heads. What is the probability that the other coin is also heads?

The standard “wrong” answer is 1/2. Supposedly, the right answer is 1/3. Here is the explanation for that “right” answer:

For two coins there are four equally probable outcomes: HH, HT, TH and TT. Obviously, TT is excluded in this case, and of the remaining three possibilities only one has two heads.

Here is the problem with this problem. Suppose I flip two coins without letting you see the outcome. If I get one head and one tail, what will I tell you? I can tell you that at least one of the coins came up heads. Or, I can tell you that at least one of the coins came up tails. The fact that I can tell you different things changes the a posteriori probabilities.

You need to base your calculation not only on your knowledge that there are only three possibilities for the outcome: HH, HT and TH, but also on the conditional probabilities of these outcomes, given what I told you. I claim that the initial problem is undefined and the answer depends on what I decide to say in each different case.

Let us consider the first of two strategies I might use:

I flip two coins. If I get two heads, I tell you that I have at least one head. If I get two tails I tell you that I have at least one tail. If I get one head and one tail, then I will tell you one of the above with equal probability.

Given that I told you that I have at least one head, what is the probability that I have two heads? I leave it to my readers to calculate it.

Suppose I follow the other strategy:

I flip two coins. If I get two tails, I say, “Oops. It didn’t work.” Otherwise, I say that I have at least one head.

Given that I told you that I have at least one head, what is the probability that I have two heads? If you calculate answers for both strategies correctly, you will have two different answers. That means the problem is not well-defined in the first place.

Teacher: Why didn’t you do your statistics homework?
Student: I read a statistical study that the students who spend more time on their homework get lower grades.
Teacher: So you didn’t do your homework in order to increase your grades?
Student: Yep.
Teacher: I have been teaching you that correlation doesn’t mean causality. Did it ever occur to you that students with good grades already know some of the material and they do not need much time to complete their homework?
Student: Oh?
Teacher: You are getting an F for not doing your homework. Now you might understand causality better.

When I came to the US, I heard about Mensa — the high IQ society. My IQ had never been tested, so I was curious. I was told that there was a special IQ test for non-English speakers and that my fresh immigrant status and lack of English knowledge was not a problem. I signed up.

There were two tests. One test had many rows of small pictures, and I had to choose the odd one out in each row. That was awful. The test was English-free, but it wasn’t culture-free. I couldn’t identify some of the pictures at all. We didn’t have such things in Russia. I remember staring at a row of tools that could as easily have been from a kitchen utensil drawer as from a garage tool box. I didn’t have a clue what they were.

But the biggest problem was that the idea of crossing the odd object out seems very strange to me in general. What is the odd object out in this list?

Cow, hen, pig, sheep.

The standard answer is supposed to be hen, as it is the only bird. But that is not the only possible correct answer. For example, pig is the only one whose meat is not kosher. And, look, sheep has five letters while the rest have three.

Thus creative people get fewer points. That means, IQ tests actually measure how standard and narrow your mind is.

The second test asked me to continue patterns. Each page had a three-by-three square of geometric objects. The bottom right corner square, however, was empty. I had to decide how to continue the pattern already established by the other eight squares by choosing from a set of objects they provided.

This test is similar to continuing a sequence. How would you continue the sequence 1,2,3,4,5,6,7,8,9? The online database of integer sequences has 1479 different sequences containing this pattern. The next number might be:

• 10, if this is the sequence of natural numbers;
• 1, if this is the sequence of the digital sums of natural numbers;
• 11, if this the sequence of palindromes;
• 0, if this is the sequence of digital products of natural numbers;
• 13, if this is the sequence of numbers such that 2 to their powers doesn’t contain 0;
• 153, if this is the sequence of numbers that are sums of fixed powers of their digits;
• 22, if this is the sequence of numbers for which the sum of digits equals the product of digits; or
• any number you want.

Usually when you are asked to continue a pattern the assumption is that you are supposed to choose the simplest way. But sometimes it is difficult to decide what the testers think the simplest way is. Can you replace the question mark with a number in the following sequence: 31, ?, 31, 30, 31, 30, 31, … You might say that the answer is 30 as the numbers alternate; or, you might say that the answer is 28 as these are the days of the month.

Towards the end of my IQ test, the patterns were becoming more and more complicated. I could have supplied several ways to continue the pattern, but my problem was that I wasn’t sure which one was considered the simplest.

When I received my results, I barely made it to Mensa. I am glad that I am a member of the society of people who value their brains. But it bugs me that I might not have been creative enough to fail their test.

I am looking forward to the 2009 Women and Math program at Princeton. The irony is that I lived in Princeton for seven years and the only time I visited this program was for the lecture course on wavelets by Ingrid Daubechies.

I felt that mathematics should be genderless and pure; that the only basis for a program should be mathematics itself. I tried to ignore the problems of women mathematicians by pretending they didn’t exist. By the time I realized that I might very well love to hang out with a large group of female mathematicians, I left Princeton.

Can you imagine how glad I was when I got a call inviting me to join the organizing committee for the Women and Mathematics Program last year? I was so eager that I arranged a math party at the Program and gave my own talk about Topology in Art.

What can I tell you? I loved the program. For the first time in my life I didn’t feel like a loner, but rather that I belonged to a group. I also felt envious, because when I was a student we didn’t have anything like this in Russia.

I am going to be on the program this year too. The subject is Geometric PDE. I am so looking forward to it that I’m already planning another math party.

Once I was at a party and a woman was complaining that her car insurance bills were enormous. Her expensive car was hit three times while it was parked. She was whining about how unfair it was for her to be paying increased insurance premiums when it hadn’t been her fault. I didn’t tell her my opinion then, but I’m going to write about it now.

Though such things can happen, it is possible to reduce the probability of your parked car being hit.

In my personal experience the most frequent parking accident happens when someone backs out of a driveway and there is a car parked in a space which is usually empty. People often back out of their driveways on autopilot. If you park on a narrow street with no other cars — a sign that people don’t usually park there, do not park across from a driveway or close to a driveway.

There are many other common sense ideas. Don’t park at a corner. Choose the better lit areas. Don’t park next to a truck or a van, because they might not see you very well and if they hit you, they’ll do more damage. Don’t park next to an old, battered car because they have less to lose than you do. New cars are the best neighbors. Not only are owners of new cars usually more careful, but new cars are also often leased. And people who lease a car are even more careful, because they have to return it in good order.

When you are choosing a perpendicular parking spot, here’s a cute idea. Pick cars with four doors as your neighbors. Cars with two doors have bigger doors and if you are too close, they might scratch you.

Here’s what I would have told that woman: If your car has been hit so many times while parked, you should rethink your parking strategy.

Browsing the Internet, I stumbled upon a coin puzzle which I slightly shrank to emphasize my point:

Carl flipped two coins and was asked if at least one of the two coins landed “heads up”. He replied, “Yes. In fact the first coin I flipped landed heads up.” What is the chance that Carl’s coins both landed heads up?

The standard answer is 1/2, because there are only two possibilities for the coin flips: HH and HT. But how do we know that these possibilities are equally probable?

The answer depends on what we expect Carl to say when he flips two heads. My personal assumption is that Carl is a perfectionist and always volunteers extra information. If Carl gets two heads, I would expect him to say, “Yes. In fact both coins I flipped landed heads up.” In this case the answer to the puzzle is 0.

Another strange but reasonable assumption is that upon flipping two heads, there is an equal probability that Carl would say either, “Yes. In fact the first coin I flipped landed heads up;” or, “Yes. In fact the second coin I flipped landed heads up.” In this case, the answer to the puzzle is 1/3.

I could describe an assumption for Carl’s answering strategy that leads to the puzzle’s answer of 1/2, but it looks too artificial to me.

This puzzle is not well-defined, but unfortunately there are many versions of it floating around the Internet with incorrect solutions.

One day I got a phone call from Victor Gutenmacher, one of the members of the jury for the USSR Math Olympiad. At that time I was 15 and had won two gold medals at the Soviet Math Olympics. Victor asked me about my math education. I explained to him that although I went to a special school for gifted children, I wasn’t doing anything else. In his opinion, other kids were using more advanced mathematics for their proofs than I was. He said I was coloring everything in black and white; other kids were using calculus, while I was only using elementary math. He asked me if I would like to learn more sophisticated mathematics.

I said, “Sure.” After considering several different options, Victor suggested Israel Gelfand’s seminar at Moscow State University. He told me that this seminar might suit me because it starts slowly, picking up pace only at the end. He also told me that the seminar was like a theater. Little did I know that I would become a part of this theater for many years to come. I also didn’t know that I would meet my third husband, Joseph Bernstein, at this seminar. Joseph used to sit in the front row, and I watched his back at the seminar for more years than I later spent together with him.

The next Monday evening, I went to the seminar for the first time. Afterwards, Gelfand approached me and asked me if I had an academic adviser. I said, “No.” He asked me how old I was. I said, “Fifteen.” He told me that I was too old and that I had to choose an adviser without delay. I said, “But I do not know anyone and, besides, I need some time to think about it.” He replied, “I’ll give you two minutes.”

I paced the halls of the 14th floor of the Moscow State University for a couple of minutes, pretending to think. But really, I didn’t know about any other options. He was the only math adviser I had ever met. So I came back and asked Gelfand, “Will you be my adviser?”

He agreed and remained my adviser until I got my PhD 14 years later.

When you talk over the phone with an adult stranger, you can generally determine if this person is male or female. From this, I conclude that the voice characteristics are often noticeably different for males and females. There are many other characteristics that have a different distribution by gender — for example, height.

My question is: “Can we find a trait such that the distributions are the same for both genders?”

Trying to find the answer, I remembered what we learned in high school about the genetics of eye color. I checked the Internet on the subject and discovered that the story is somewhat more complicated than what I studied 30 years ago, but still we can say that eye color is defined by several genes, which are located on non-sex chromosomes. That means, your eye color depends on the genes your parents have and doesn’t depend on your sex. A boy and a girl from the same parents have the same chances for any particular eye color.

Since eye coloring has nothing to do with gender, women and men are equal in the eyes of eye colors.

Does that mean that if we check the distribution of eye color for the world population, the distribution histogram will be the same for men and women? That sounds like a logical conclusion, right? I would argue that this is not necessarily the case.

Let me remind you that the distribution of eye color depends on the country. China has an unprecedented gender imbalance, with 6% more men than women in its population. As the eye color of Chinese people is mostly dark brown, this creates an extra pool for a randomly chosen man in the world to have a darker eye color than for a randomly chosen woman. If we exclude China from consideration, we can still have different distributions. For example, in Russia the life expectancy for women is 15 years longer than the life expectancy for men. Consequently, Russia has 14% less men than women, while globally the male/female sex ratio is 1.01. Therefore, eye colors common in Russia will contribute to female eye colors more than those of male.

What if we consider only one country? Let us look at the US. Immigrants to the US are mostly males. If the distribution of eye color for immigrants is different than the distribution for non-immigrants, then male immigrants contribute more to the eye color distribution than female immigrants.

There are so many factors impacting eye color distribution, that it isn’t clear whether it’s possible to find a group of people other than siblings in which the distribution of eye color would be the same for women and men.

We see that eye color distribution, which theoretically doesn’t depend on gender, when measured in a large population can produce different distributions for men and women.

Recently I wrote a theoretical essay titled “Math Career Predictor”, where I assumed that the distribution of math ability is different for men and women. In reality, there is no good way to measure math ability, hence we do not have enough data to draw a complete picture. For the purposes of this discussion let us assume that we can measure the math ability and that Nature is fair and gave girls and boys the same math ability. My example with eye color shows that if we start measuring we might still see different distributions in math ability in boys and girls.

My conclusion is that if we measure some ability and the distribution is different for boys and girls, or for any other groups for that matter, we can’t just conclude that boys and girls are different in that ability. For some distributions, like voice, we probably can prove that the difference is significant, but for other characteristics, different distribution graphs are not enough; we need to understand the bigger picture before drawing conclusions.