Tanya Khovanova's Math Blog

I once wrote a story about a mistake that my medical insurance CIGNA made. They had a typo in the year of the end date of my insurance coverage in their system. As a result of this error, they mistakenly thought they had paid my doctors after my insurance had expired and tried to get their money back. While I was trying to correct all this mess, an interesting thing happened.

To help me explain, check out the following portion of my bill. (If it looks a bit funny, it’s because I cut out some details including the doctor’s name).

On the bill you can see that I had a mammography for which I was charged $493.00, but CIGNA paid only $295.80. The remaining $197.20 was removed from the bill as an adjustment, as frequently happens because of certain agreements between doctors and insurance companies. A year later when CIGNA made their mistake, they requested that the payment be returned. You can see on the bill that once the payment was reversed, my doctors reversed the adjustment too.

When CIGNA fixed the typo, they repaid the doctors, but the adjustment stayed on the bill, which the doctors then wanted me to pay. And that was only one of many such bills. It took me a year of phone calls to get the adjustments taken off, but this is not what I am writing about today.

If not for this mistake, I would have never seen these bills and the revealing information on the different amounts doctors charge to different parties, and how much they really expect to receive. As you can see my doctors wanted 67% more for my mammogram than they later agreed to.

The difference in numbers for my blood test was even more impressive. I was charged $173.00, and the insurance company paid $30.28 — almost six times less.

If I ever need a doctor and I don’t have insurance, I will take these bills with me to support my request for a discount. I do not mind if you use this article for the same purpose.

Dear Nita Palmira,

I do not recall your name and I’m not sure where you got my email address from, but I really appreciate you contacting me. I am excited by your Two-Procedures-For-The-Price-Of-One offer. I am really looking forward to my enlarged penis and my DDD breasts.

Meanwhile, I can give you a unique group discount on IQ tests. I can test the IQ of all your company employees for the price of one test. Moreover, you do not need to waste even a minute. Actually, no one even needs to answer any questions. You can send me your $500 check to the address below and I will promptly send you the IQ report, the accuracy of which I can guarantee.

Sincerely yours.

Credit cards often keep your mother’s maiden name in their database for security purposes. This so called “security” is based on two assumptions:

• Your mother was married.
• Your mother changed her last name to her husband’s last name.

Were these assumptions true, only your close relatives would know your mother’s maiden name. In reality, if your mother was never married, then your last name is the same as your mother’s last name. So, crooks who are trying to steal identities can try to use your last name as your mother’s presumed maiden name. Very often they will succeed. Besides, many women do not change their last names. If you have a different last name from your mother, but your mother uses her maiden name, then the bank’s security question is not very secure at all.

If you want your identity to be secure you might need to invent a maiden name for your mother. Alternatively, perhaps your parents can tell you a family secret that will help you choose a name that is related to you, but not transparent to the public.

My relative Martin took his wife’s last name after their marriage. Before his children apply for credits cards and bank accounts, he needs to explain to them that it is better for them to use his maiden name as their mother’s maiden name for banking purposes.

I love Raymond Smullyan’s books , especially the trick puzzles he includes. The first time I met him in person, he played a trick on me.

This happened at the Gathering for Gardner 8. We were introduced and then later that day, the conference participants were treated to a dinner event that included a magic show. In one evening I saw more close-up magic tricks than I had in my whole life. This left me lightheaded, doubting physics and my whole scientific outlook on life.

Afterwards, Raymond Smullyan joined me in the elevator. “Do you want to see a magic trick?” he asked. “I bet I can kiss you without touching you.” I was caught off guard. At that moment I believed anything was possible. I agreed to the bet.

He asked me to close my eyes, kissed me on the cheek and laughed, “I lost.”

Suppose you want to increase your chances of winning the lottery jackpot by pooling money with a group of coworkers. There are several issues you should keep in mind.

When you pool the money and you hit the jackpot, the money has to be split. If you bought 10,000 tickets and the jackpot that you win is $100 million, then each ticket is entitled to a mere $10,000. Your chances of hitting the jackpot in the first place are 1 in 17,500 and you’re not going to get rich off what you win.

Perhaps you’d be satisfied with a small profit. However, as I calculated in my previous piece on the subject, even if you include the jackpot in the calculation of the expected return, the Mega Millions game never had, and probably never will have a positive return.

Despite this fact, people continue to pool money in the hopes of winning big. However, there are more problems in doing this than just its non-profitability.

Consider a scenario. Your coworkers collected $1,000 to buy 1,000 lottery tickets. You give the money to Jerry who buys the tickets. Jerry can go to a store and buy 1,005 tickets. After the lottery he checks the tickets, takes the best five for himself and comes back to work with 1,000 disappointing tickets.

It is more likely that Jerry is cheating or that he will lose the tickets than it is that your group will win the jackpot. But there is a probabilistic way to check Jerry’s integrity. According to the odds, every 40th ticket in Mega Millions wins something. Out of 1,000 tickets that Jerry bought, you should have about 25 that win something. If Jerry systematically brings back tickets that win less often than expected, you should replace Jerry with someone else.

There are methods to protect your group against cheating. For example, you can ask another person to join Jerry in purchasing the tickets, which they then seal in an envelope that they both sign.

Alternatively, you yourself could be the person responsible for buying 1,000 tickets. How would you protect yourself from suspicion of cheating? The same way as I mentioned above: bring along some witnesses and have everyone sign the sealed envelope.

The most reliable way to prevent Jerry from cheating is to have him write down all the ticket numbers and send this information to everyone before the drawing. This way he can’t replace one ticket with another. But this is a lot of work for tickets that are usually worth less than the money you collected to buy them.

But there are other kinds of dangers if you use this supposedly reliable method. If you bought a lot of tickets the probability of winning a big payoff increases. Suppose Jerry publicly locks the envelope in a desk drawer in his office. If one ticket wins $10,000, and everyone knows all the ticket combinations, suddenly Jerry’s desk drawer becomes a very unsafe place to keep the tickets.

Scams are not your only worry. You shouldn’t buy the same combination twice — whether picking randomly or not. You really do not want to waste a ticket and end up sharing the jackpot with yourself.

You cannot change the odds of hitting the jackpot, but you can change the odds of sharing it with others. Indeed, there are people who do not buy random combinations, but rather pick their favorite numbers, like birthdays. You can reduce the probability of sharing the jackpot if you choose the combinations for your tickets wisely, by picking numbers that other people are unlikely to pick.

Still want to try the lottery? If you feel a need to throw your money away, instead of buying lottery tickets, feel free to donate to this 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.

To translate from a Russian joke, borrowing money is taking someone else’s: temporary; giving back your own: forever.

This is a story about my great-uncle Fred. His name is not Fred, of course, because I don’t want to reveal which one of my thirteen great-uncles created this ingenious scheme.

My great-uncle Fred asked to borrow 100 rubles from my mother. He was notorious for not returning money, but he knew how to work my mother. He whined about being sick and urgently needing to buy pills, until my mother, who has a big heart and is an easy touch, gave up. Of course, Fred wasn’t in a hurry to return the money. But 100 rubles was a lot of money for my mother and she wasn’t planning on giving up trying to get it back. My mother started bugging her uncle with increased intensity. Finally Fred promised to return the money as a gift for mom’s upcoming birthday.

Of course, it was tacky to present the money he owed as a gift, but my mom was so glad that she would finally get her money back, that she was actually looking forward to it.

During her party, as the guests sat around the table, Fred got up to give the birthday toast. Then Fred handed my mother an envelope and said, “Congratulations on your birthday! Here is a gift for you.” Everyone applauded.

My mom felt that something in this scene was not quite right. Why was the applause so enthusiastic when he was just returning a debt? After the party my mother decided to investigate. It turned out that Fred explained to my mother’s relatives that she prefered money as a birthday gift and collected the gift money from everyone. The cash he returned as his debt in the envelope was not his. Everyone else thought he was presenting the joint gift, except for my mother, who was made to believe that he was repaying his debt.

After that my mother stopped bugging her uncle Fred. It became clear she couldn’t match his superb skills in escaping his debts.

Here is an arithmetic problem for you:

You have $700 dollars in your checking account. You are sloppy and forget how much you have. You write three checks for $600, $200 and $200. For every bounced check you are fined $25 by your bank. How much in fines will you have to pay for your sloppiness?

Solution: the fine depends on the transaction order. If they process your $600 check first, you will have two bounced checks. If they process a $200 check first, then only your $600 check will bounce.

The question is, what will your bank do if all three checks need to be processed at the same time? There are three options:

• Your bank doesn’t have a good mathematician on the staff and is not aware of this situation, and it processes the checks in random order. In this case you will have either two bounced checks (with a probability of 1/3) or one bounced check (with a probability of 2/3).
• Your bank is evil, and purposefully processes your $600 check first. In this case you are guaranteed to have two bounced checks.
• Your bank cares about its soul and purposefully processes the $600 check last. In this case you are guaranteed to get only one bounced check.

Assuming the worst — your bank is evil — what is the answer to the problem? Do you think you will be fined $50? If so, you are wrong. The company to whom you wrote the check will fine you too. Supposing that the company has the same $25 fine as the bank, can we say that you will be fined $100? Nope, this is not correct either. You are forgetting that companies will reprocess your bounced checks two days later and the checks will bounce again. You will be fined twice for each check by two different entities. Thus, you can face $200 in fines.

My next question is: what do you think is a fair fine in my arithmetic problem above?

Banks and companies have never heard of double jeopardy and do not think that it is unconstitutional to fine you twice for the same mistake. No doubt, the second reprocessing of your checks is done “for your convenience”. “For your convenience” they assume that the bouncing was due to a computer glitch, so they should reprocess your check immediately after it has bounced. “For your convenience” no-one will disturb you to notify you that your checks are bouncing. I also believe that if your fine depends on the random order of processing of checks, the banks should be graceful and shouldn’t pick the more profitable order for themselves. I do think that charging you more than $50 in my example is against the law and is not fair.

The law should protect us against entities that rob us “for our convenience.”

Start is the STate of the ART question-answering system. You can ask Start any question in plain English — for example, “What is the population of Moscow?” — and instead of producing millions of pages like Google, it provides one exact answer: “The population of Moscow, Russia, is 8,746,700.” I am not sure where this number comes from, as Russian sites suggest that the population of Moscow is more than 10 million people. But anyway, back to my challenge.

I have my email address in plain sight on my webpage. As a result, I get a lot of spam. So, I am thinking about a way to present my address so that humans can easily deduce it, but computers can’t. Here it is: my email server is Yahoo and my user name consists of 7 lower case letters. Each letter answers one of the questions below, in the right order. As of today, Start can’t answer any of these questions.

1. What is the first letter of the word 3?
2. What is the first letter of the alphabet?
3. What is the only common letter in the words “knowledge” and “triamphant”?
4. What is the last letter of all the days of the week?
5. What is the first letter of almost all the continents?
6. What is the first letter of the word “knight”?
7. What is the most frequent letter in the word “although”?

The advantage of presenting my user name in this manner is that I will restrict my new correspondence to people who are sufficiently eager to write to me that they can spare ten seconds figuring out my email address. The main advantage is that Start can’t answer these questions, giving me hope that spamming software can’t do it either.

I do think that the state of the art question-answering system should know the first letter of the alphabet. Start: these questions are a challenge for you. How much time will it take you to do it?

Watch out. Maybe Google can do it faster.

I was waiting for a train in Newark. On the platform, there was an LCD screen that flashed advertising. The ad I was staring at was titled, “Reasons to take NJ Transit to Prudential Center, #15.” I was impressed that the NJ Transit sales people were working so hard to invent that many reasons.

I waited for my train for half an hour. It turned out that the NJ Transit advertising people were not working hard after all. The screen was flipping between four reasons, numbered 3, 6, 12 and 15. This is a case of false advertising. You look at reason number 15 and think that there must be a lot of reasons. They fool with your head. Cheaters.

I hope you noticed that I did the same thing with this posting — purely in order to illustrate my point.