Tanya Khovanova's Math Blog

Sara was born in Boston on February 29, 2008 at 11:00 am. Her parents were quite upset that their calendar-challenged daughter would only be able to celebrate her birthday once in four years. Luckily, science can help Sara’s parents. How? Sara can celebrate her birthday every year at the moment when the Earth passes the same point on its orbit around the Sun as when Sara was born.

Assuming that Sara lives her entire life in Boston and that the daylight savings time is not moved earlier into February, your task is to calculate the schedule of Sara’s birthday celebrations for 100 years starting from her birth. To simplify your homework, you can approximate one year as 365 days and 6 hours.

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.

In one of my previous pieces, I discussed returns on the Mega Millions lottery game, assuming that you buy a small number of tickets. In such a case winning the jackpot has zero probability. So I argued that if you want to estimate the profitability of the lottery as an investment, you have to remove the jackpot money from the calculation.

Today I will discuss what the formal expected return is. That is, I will include the jackpot money in the calculation. Since I argued against including the jackpot in my last article, you might wonder why I’ve then turned around to look into this.

I think this mathematical exercise will be fun. Besides, on a practical note, it is useful to know when the formal expected return is more than 100%, because then it might make sense to pool money with other people. Keep in mind though that if you want a chance to hit the jackpot, the total number of tickets you buy must be really big. For example, even if you manage to pool $10,000 for tickets, your probability of winning the jackpot in Mega Millions is only one in 17,500 — still minuscule.

If you buy only one ticket, you’ll lose. If you manage to pool a lot of money and the probability of the jackpot becomes noticeable, that is, non-zero, could the jackpot be large enough that the lottery becomes a good investment?

For this calculation, I’m still assuming that you buy a relatively small number of tickets. If you buy millions of tickets the calculation is slightly different, and I will write about that later.

You might think that when the jackpot is bigger than the odds, it makes sense to play. I am discussing the Mega Millions game, where the odds of winning the jackpot are one in 175 million. So if the jackpot is more than $175 million, then it is profitable to play. Right?

Wrong. As I mentioned in my previous piece, after reducing for taxes, you get about 16% of your money back through smaller payouts. Hence, you need to recover the other 84% through the jackpot. So the jackpot should be more than 175*.84 = 147 million dollars. This sounds even better. Right?

Wrong. No one receives the jackpot. Winners can chose to immediately receive the lump sum, which equals the money lottery organizers have actually set aside for it. Alternatively, the lottery organizers can invest the lump sum and give winners a yearly distribution over many years, the total of which will equal the jackpot.

Suppose for simplicity the lump sum is half of the jackpot. That means we need the jackpot to be $294 million ($147 x 2). Right?

Oops. As usual, we forgot about taxes. To exacerbate your pain, I have to add that the winnings are taxable. Suppose you have to pay 30% from the jackpot. That means the jackpot needs to be $424 ($294/0.7) million in order to justify pooling money. OK?

We haven’t seen jackpots that big yet. But neither have we finished the calculation. There is a probability that you might have to share the jackpot with other winners. To calculate this probability, we need to calculate the number of tickets sold. That means, your expected return depends not only on the size of the jackpot, but also on the number of these tickets.

But even if you know the number of tickets sold, we cannot calculate the expected returns precisely because people don’t always buy tickets with random combinations, but often pick their own numbers.

When the jackpot is large people start buying tons of tickets, so we can expect that many of them buy quick-picks. Let us assume for now that the vast majority of people do not choose their own numbers, but buy tickets at random. Suppose 200 million tickets were sold. That is a very big number. Last time that many tickets were sold was when the jackpot was $390 million in March 2007. By the way, that was the largest jackpot ever.

In order to finish the calculation, we need to establish the probability of several winners, given that 200 million random tickets were sold:

From here we can calculate the adjustment coefficient, that is, the proportion of money you are expected to get from the jackpot given that there are 200 million players in the game. The coefficient is calculated from the table above as (0.3647 + 1/2*0.2075 + 1/3*0.0787 +1/4*0.0224 + 1/5*0.0051 + 1/6*0.0010)/(1 – 0.3204), and is equal to 0.7379. We need to divide our previous figure of $424 million by the adjustment coefficient. The result is $575 million.

Given that a $390 million jackpot attracted more than $200 million in tickets, we can expect that the $575 million jackpot will make people completely crazy and attract even more money. So I do not anticipate that the Mega Millions game will ever have a positive formal expected gain. My conclusion is that not only is there no financial sense in buying a single lottery ticket, but also none in pooling money.

Of course, you can buy tickets for non-financial reasons, like pumping up your adrenaline. In any case, I showed you the method to calculate your expected return, or, more appropriately, your expected loss.

Lottery is a tax on people bad at math.

In this article I calculate how bad the lottery is as an investment, using Mega Millions as an example. To play the game, a player pays $1.00 and picks five numbers from 1 to 56 (white balls) and one additional number from 1 to 46 (the Mega Ball number, a yellow ball).

During the drawing, five white balls out of 56 are picked randomly, and, likewise, one yellow ball out of 46 is also picked independently at random. The winnings depend on how many numbers out of the ones that a player picks coincide with the numbers on the balls that have been drawn.

So what is your expected gain if you buy a ticket? We know that only half of the money goes to payouts. Can you conclude that your return is 50%?

The answer is no. The mathematical expectation of every game is different. It depends on the jackpot and the number of players. The more players, the bigger is the probability that the jackpot will be split.

Every Mega Millions playslip has odds printed on the back side. The odds of hitting the jackpot are 1 in 175,711,536. This number is easy to calculate: it is (56 choose 5) times 46.

How much is 175,711,536? Let’s try a comparison. The government estimates that in the US we have 1.3 deaths per 100 million vehicle miles. If you drive one mile to buy a ticket and one mile back, your probability to die is 2.6/100,000,000. The probability of dying in a car accident while you drive one mile to buy a lottery ticket is five times higher than the probability of winning the jackpot.

Suppose you buy 100 tickets twice a week. That is, you spend $10,000 a year. You will need to live for 1,000 years in order to make your chances of winning the jackpot be one out of 10. For all practical purposes, the chance of winning the jackpot are zero.

As the probability of winning the jackpot is zero, we do not need to include it in our estimate of the expected return. If you count all other payouts then you are likely to get back 18 cents for every dollar you invest. You are guaranteed to lose 82% of your money. If you spend $1000 a year on lottery tickets, on average you will lose $820 every year.

If you do not buy a lot of tickets your probability of a big win is close to zero. For example, the probability of winning $250,000 (that is guessing all white balls, and not guessing a yellow ball) by buying one ticket is about 1 in 4 million. The probability of winning $10,000 — the next largest win — is close to 1 in 700,000. If we say that you have no chance at these winnings anyway, then your expected return is even less: it is 10 cents per every dollar you invest.

You might ask what happens if we pool our money together. When a lot of tickets are bought then the probability of winning the jackpot stops being zero. I will write about this topic later. For now this is what I would like you to remember. From every dollar ticket:

• 50 cents goes to the state
• 32 cents towards the jackpot
• 18 cents to other winners

I am not at all trying to persuade you not to buy tickets. Lottery tickets have some entertainment value: they allow you to briefly dream about what you would do with those millions of dollars. But I am trying to persuade you not to buy lottery as an investment and not to put more hope into it than it deserves. If you treat lottery tickets as tickets to a movie that is played in your head, you will never buy more than one ticket at a time.

That is it. I advise you not to buy more than one ticket at a time. One ticket will allow you to dream about the expression on your sister’s face when she sees your new $5,000,000 mansion, but will not destroy your finances.

An ancient Russian joke:

Patient: Doctor, is there a medicine I can use to prevent my girlfriends from become pregnant?
Doctor: Kefir.
Patient: Should I drink it before or after sex?
Doctor: Instead of.

I have a more pleasurable suggestion than drinking kefir: date postmenopausal women. There are many other reasons why men enjoy dating older women, but since my blog is about mathematics, I would like to dig into some relevant numbers.

We know that boys are born more often than girls, and men die earlier than women. Somewhere around age 30 the proportion in population switches from more boys to more girls. And it gets more skewed with age. So there’s a deficit of older men. In addition, a big part of the population is married, making the disproportions in singles group more pronounced. So I decided to look at the numbers to see how misshaped the dating scene is.

This 2008 data comes from the U.S. government census website’s table “Marital Status of the Population by Sex and Age: 2008. (Numbers in thousands. Civilian non-institutionalized population.)” To calculate the number of singles, I summed up the widowed, divorced and never married columns.

These data alone cannot explain the dating situation. For example, I have no way of knowing what proportion of each gender isn’t interested in dating the opposite sex, or even in dating altogether. But the trend is quite clear: the proportion of men in younger categories is much higher. That implies that there is less competition for older women. So those young men who are open to dating much older women might have more options and those options might be more interesting.

I just turned 50 and plan to return to dating again. Looking at the data I see that there are 11 million single men older than me and 34 million who are younger than me. If I were to pick a single man randomly, I am three times more likely to end up with a younger man.

Supposedly we live in a free society, where people can do what they want as long as they do not harm anyone else. Still our society often disapproves of women dating much younger men. Consider this definition from Wikipedia:

“Cougar — a woman over 40 who sexually pursues a much younger men.”

This derogatory term portrays such women as predatory. Not only is there nothing wrong with women dating younger men, but it makes no sense for older women to ignore the imbalance of the dating scene and be closed to relationships with much younger men. After all, the demographics are also affected by the fact that women live longer, probably because of their healthy life style, non-risky behavior and positive attitude to life.

Can someone explain to me again why sane, healthy, non-risky women with positive attitudes to life are called “cougars”?

I remember this question from my childhood:

Why is the South Pole colder than the North Pole?

Indeed, the average winter temperature at the North Pole of -34°C is the same as the temperature at the South Pole at the beginning and end of its summer. The South Pole is only warmer than the North Pole 40 days per year. So the South Pole is a much, much colder place. According to Wikipedia there are three major reasons for this:

1. The North Pole is at sea level, while the South Pole is elevated to almost three kilometers. The higher a land mass the colder it is.
2. The North Pole sits on water whose temperature never goes below -2°C. Compared to the South Pole, this is like keeping the North Pole on the stove top.
3. The South Pole is farther from the ocean, so it has higher continentality, which is usually associated with colder temperatures.

I remember when I was a child my father gave me a completely different explanation.

The Earth’s orbit is not a circle, but rather an ellipse. According to Kepler’s second law: “A line joining a planet and the sun sweeps out equal areas during equal intervals of time.” This means that the earth has a slower angle motion around the aphelion — in its furthest point — than around the perihelion — in its closest point to the Sun. Consequently, the summer is longer than the winter for the North Pole, whereas the opposite is true for the South Pole.

Something in my father’s explanation bothered me. Now I understand what: though the summer is longer at the North Pole, it should get less sunshine as the North Pole is further away from the Sun than the South Pole during its summer. So the effects might cancel each other out. In any case, as the earth’s orbit is almost circular, the contribution of the shape of the orbit should be minor, compared to the effects of elevation, the water underneath and continentality.

On the other hand, it is possible that my father wasn’t talking about the poles, but rather about the difference in hemispheres. I wonder if someone can calculate if there is a difference in the amount of sunlight the poles get due to the fact that the Earth’s orbit is not circular. Is the temperature different for the places that are equidistant from the equator, and have similar elevation and continentality, but which are located in different hemispheres?

I remember a funny article explaining why the northern hemisphere has more land. They said that continents drifted into the northern hemisphere because they wanted a nicer climate.

Mathematically we can describe a marriage by a graph. People are vertices and two spouses are connected by an edge.

Mathematical models tend to oversimplify life, so let us assume that a person can only be one of two genders. Therefore, the vertices of a graph are colored in two colors: pink and blue. In this article I explore the graph theory of different types of marriages.

A monogamous couple is represented as a complete K₂ graph: two vertices connected by an edge. The graph is bipartite, no matter how you color it. But actually our vertices are already colored from the start. If we are considering traditional marriage, one vertex is pink and the other is blue.

Historically, the second most common type of marriage is polygyny, in which one man has several wives. Less common in history, but a mathematically equivalent type, is polyandry, in which one woman has several husbands. Both these types of marriages emphasize inequality, as husbands and wives have completely different sets of rights.

From a mathematical point of view, polygyny and polyandry are described by star graphs. Star graphs are bipartite graphs and the natural coloring is the one that proves bipartiteness.

The final type of marriage is polygynandry, which refers to a group marriage, where more than one man and more than one woman create a family. Everyone can have sex with everyone else of the other gender. Mathematically this type of marriage corresponds to a complete bipartite graph K_n,m. Actually, in this case I can imagine that a particular pair of people of different genders wouldn’t like each other and might not consummate their marriage. So this graph is not necessarily complete.

How can same-sex marriages change the graph theory of marriages? As a graph, a monogamous same-sex marriage is the same bipartite K₂ graph as a heterosexual marriage. It will just be less colorful, as both vertices will be of the same color.

But what happens if we add the same-sex idea to polygamous marriages?

Suppose a homosexual man wants to live with several spouses at the same time. What name can we give to a family unit of more than two homosexual men? Homopolygamy? Their marriage graph will be a star graph in which all the vertices are of the same color.

If a man can have several spouses, what about his spouses? Can they form multiple marriages too? If only one person is allowed to engage in several marriages, then we will see inequality within the same gender. If any spouse is allowed to form other marriages, then we will have a situation in which several men are all spouses to each other. So mathematically we will see complete graphs with more than two vertices to represent a marriage. If two people in a group do not like each other and do not want to be married, then the corresponding graph doesn’t need to be complete.

By symmetry we can describe a marriage of several women, and mathematically it will be similar to a marriage of several men.

Another interesting aspect is the idea of mixed types of marriages involved in polygamy. Suppose a husband has several wives. Some of them might get bored waiting for his attention, and start spending so much time with each other that they end up developing feelings for each other. Suppose two wives of the same man decide to marry each other. What name would we give to this type of marriage? I am afraid that we do not have enough words to cover all the potential situations.

Suppose we have a heterosexual married couple and the man decides to bring another woman into their house. Thus the transition from a traditional marriage to a polygyny is created. If they got along so well that the first wife decides to marry the second wife, this would require a transition to a new type of marriage. Oh, I see that my essay just went in another direction — how different types of marriages might evolve into each other. For now, I’ll leave this for future research.

Talking about different directions. I recently wrote a piece about condoms. Now I have a new generalization for the classic condom puzzle. Suppose we have a mixed-type marriage defined by a graph. Suppose tonight every couple of people corresponding to the edge of this graph wants to have sex with each other. What is the smallest number of condoms they can use? In my condom essay, I didn’t define the condom usage for the sex of two women. I will leave it to your imagination and definition.

My son Alexey taught me to play “The Settlers of Catan .” This game is so good that throughout the four years of his undergraduate studies, he played it every evening. I am exaggerating of course, but only so slightly. He also taught me some of the game’s wisdom.

Lesson 1. Trading is beneficial for both traders.

When you agree to exchange your two rocks for one grain, one grain is more valuable to you than two rocks. The opposite is true for your trading partner.

Presumably, the same principle works for the economy. If I buy a sweater at T.J.Maxx for $20, I need the sweater more than $20. And if the store sells this sweater for $20, they are hoping to make some profit, that is, that the sweater cost them less than $20. Supposedly, shopping transactions are profitable for both parties.

Lesson 2. Trading is bad for non-trading players.

This is the consequence of the fact that in “Settlers of Catan” there is only one winner. If something is good for someone, it is bad for everyone else. In real life you do not have to lose if someone wins. With each shopping transaction everyone gains. This is the reason why shopping must be good for the economy.

Lesson 3. Powerful players can persuade other players to trade against their best interests.

Shortly after I moved to the US, I became very aware of my own smell. My smell didn’t change with my move from Russia, nor did my sense of smell change. I was just bombarded with deodorant advertisements, and due to the vulnerability of my self-perception, in one year I bought more deodorants than in all my previous 30 years. I have a friend who has an exceptional sense of smell. He told me that people often use much more deodorant and perfume than they need.

Lesson 4. You pay a lot for storage.

In Settlers, if you have more than seven cards and the dice rolls seven, you need to discard half of your hand. So if you have six cards and someone offers you three grains for one sheep, consider the storage price before jumping into this bargain deal.

Once I bought so much discounted toilet paper that it lasted me for months and months. When it was time to move to a different apartment, I had to pay for the largest truck available to fit all my junk.

Lesson 5. It is important to understand the goal of the person you are trading with.

A profitable deal becomes a big mistake when, as a result of the trade, your trading partner builds a settlement right in the spot where you were planning to build.

Similarly, if your doctor prescribes you a medication, it would behoove you to know whether he will reap any profit from it himself.

Lesson 6. If a player is the only receiver of rock in the game he dictates the price.

This is like a monopoly. I needed my last laptop more than the $1,000 I paid for it. But this price included pre-installed Windows, which I didn’t want and which I immediately deleted. I was forced to pay extra for Windows because of Microsoft’s monopoly.

So, is shopping good for the economy?

What about that skirt I bought and never used and eventually threw away? I wasted $20 on it. But the store didn’t gain that $20; they only gained their profit margin, which could have been $5. That means that together we wasted $15.

I do not throw away every piece of clothing I buy, but it is true that we buy more things than we need.

I think that going shopping to help our country get out of an economic crisis is a ridiculous idea. If you are shopping for other reasons than necessity, you do not help anyone and as a group we lose.

My son Alexey wins almost every game of “Settlers of Catan” he plays. So does my friend Mark Shiffer. The main reason is that they both know how to use trading effectively. To me that indicates that there are probably other people out there who know how to effectively sell deodorants, pills, clothing and other junk to us. I suspect that I lose in every shopping transaction, as I am an unskilled trader. If most folks are like me, could it be that shopping is actually bad for the economy?

John H. Conway is teaching me his doomsday algorithm to calculate the day of the week for any day. The first lesson was devoted to 2009. “The 2009’s Doomsday is Saturday” is a magic phrase I need to remember.

The doomsday of a particular year is the day of the week on which the last day of February falls. February 28 of 2009 is Saturday, thus 2009’s doomsday is Saturday. For leap years it is the day of the week of February 29. We can combine the rules for leap years and non-leap years into one common rule: that the doomsday of a particular year is the day of the week of March 0.

If you know the day of the week of one of the days in 2009, you can theoretically calculate the day of the week of any other day that year. To save yourself time, you can learn by heart all the days of the year that fall on doomsday. That is actually what Conway does, and that is why he is so fast with calculations. The beauty of the algorithm is that the days of the doomsday are almost the same each year. They are the same for all months other than January and February; and in January and February you need to make a small adjustment for a leap year. That gives me hope that after I learn how to calculate days in 2009 I can easily move to any year.

To get us going we do not need to remember all the doomsday days in 2009. It is enough to remember one day for each month. We already know one for February, which works for March too. As there are 28 days in February, January 31 happens on a doomsday. Or January 32 for leap years.

Now we need to choose days for other months that are on doomsday and at the same time are easy to remember. Here is a nice set: 4/4, 6/6, 8/8. 10/10. For even months the days that are the same as the month will work. The reason it works so nicely is that two consecutive months starting with an even-numbered month, excluding February and December, have the sum of days equaling 61. Hence, those two months plus two days are 63, which is divisible by 7.

Remembering one of the doomsdays for every other month might be enough to significantly simplify calculations. But if you want a day for every month, there are additional doomsday days to remember on odd numbered months: 5/9, 9/5, 7/11 and 11/7. These days can be memorized as a mnemonic “9-5 job at 7-11,” or, if you prefer, “I do not want to have a 9-5 job at 7-11.”

If you throw in March 7, then the rule will fit into a poem John recited to me:

The last of Feb., or of Jan. will do
(Except that in leap years it’s Jan. 32).
Then for even months use the month’s own day,
And for odd ones add 4, or take it away*.

*According to length or simply remember,
you only subtract for September or November.

Let’s see how I calculate the day of the week for my friend’s birthday, July 29. The 11th of July falls on the doomsday, hence July 25 must be a doomsday. So we can see that my friend will celebrate on Wednesday this year.

You might ask why I described this trivial example in such detail. The reason is that you might be tempted to subtract 11 from 29, getting 18 and saying that you need to add four days to Saturday. In the method I described the calculation is equivalent, but as a bonus you calculate another day for the doomsday and consequently, you are getting closer to John Conway who remembers all doomsdays.

My homework is the same as your homework: practice calculating the days of the week for 2009.

Last time Sue refinanced her mortgage was six years ago. She received a 15-year fixed loan with 5.5% interest. Her monthly payment is $880, and Sue currently owes $38,000.

Sue is considering refinancing. She has been offered a 5-year fixed loan with 4.25% interest. You can check an online mortgage calculator and see that on a loan of $38,000, her monthly payments will be $700. The closing costs are $1,400. Should Sue refinance?

Seems like a no-brainer. The closing costs will be recovered in less than a year, and then the new mortgage payments will be pleasantly smaller than the old ones. In addition, the new mortgage will last five years instead of the nine years left on the old mortgage.

What is wrong with this solution? What fact about Sue’s old mortgage did I wickedly neglect to mention? You need to figure that out before you decide whether Sue should really refinance.