God Plays Dice

When Businesses Can Do Math, from Grey Matters -- interesting links about companies that can do math and use it to rip people off. Check out the fine print to this CashCall.com ad with Gary Coleman:

The APR for a typical loan of 2,600ドル is 99.25% with 42 monthly payments of 216ドル.55

Yes, you read that right. The interest rate is almost ONE HUNDRED PERCENT. A person taking out such a loan will end up paying back a total of 9,095ドル.10. Their commercial makes it sound like they're lending money because they "trust" people, but that's certainly not the case. I suspect that at least one of the two following things is true:

• The people taking out these loans have a very high default rate. And I mean very high; if they were giving these loans at 30% (which is a typical rate for people with credit cards who have made a ot of late payments) then the monthly payment would be 100ドル.69 over 42 months (see this calculator), for a total amount paid of 4,228ドル.98. So I'm inclined to assume that the percentage of people who pay their loans back is less than half of the percentage who pay their credit cards back.


• There's not much competition for this sort of loan; people actually see this commercial and make a phone call without bothering to shop around. This seems pretty reasonable, because the sort of people who would shop around for the best deal are probably less likely to get themselves in this sort of trouble in the first place. Therefore, the loan companies charge the highest interest rate they can legally get away with. In fact, it wouldn't surprise me to learn that in the state where these

(Incidentally, I was looking for something to calculate the payments on a loan for me; the first google hit; I wanted to check the payments on the CashCall.com loan. It complains that 99.25 is not in the range from 0.01 to 99 and so I couldn't possibly have meant that interest rate. I did!

Here are CashCall.com's rates; it seems that in states where larger loans are offered, such as California, the interest rates on those larger loans are lower (as low as 21%). Also, for some reason they only offer loans in the amounts of 1,000,ドル 2,000,ドル 2,600,ドル 5,075,ドル 10,000,ドル and 20,000ドル; does anybody have any idea why?

As for the high interest rates, I've heard that small payday loans -- say, the type where someone borrows 100ドル and has to pay back 115ドル a couple weeks later -- have to have high interest rates because the cost of doing all the administrative work for the loan needs to be covered. But it seems a lot harder to believe this on loans such as those given by CashCall.com.

This reminds me of the Comcast "Service Protection Plan" I wrote about a few days ago, which I concluded was a ripoff. I was telling my father about this today, and he pointed out that "if they're trying to sell it to you, they expect to make money off of it". The difference is that Comcast was talking about a few dollars a month, whereas payday lenders are giving people a way to really trash their financial lives.

About a month ago, I wrote a post about two semi-fictional sports teams called the Phillies and the Qankees. I asked: what is the probability that the Phillies win a best-of-seven series of games, given that they win each game with probability p? And how does this compare to the probability of them winning each individual game? It turned out that if we had p = 1/2 + ε, then the probability of them winning the series was about 1/2 + (35/16)ε (assuming ε was small). In general, the probability of winning a best-of-(2n-1) series (that is, a series that ends when the best team wins seven games) was about 1/2 + (1.11 n^1/2) ε; I conjectured that the constant 1.11 was actually 2/√π.

We can easily compute the average number of games that are actually played in a seven-game series. If we let p be the probability that the Phillies win any given game, and q be the probability that the Qankees (now do you see why I called them the Qankees?), then the number of games we expect to be played is just

4(q⁴ + p⁴) + 5(4q⁴p + 4p⁴q) + 6(10q⁴p² + 10p⁴q²) + 7(20q⁴p³ + 20p⁴q³)

since q⁴ + p⁴ is the probability that the series goes 4 games, and so on. If, as in the previous post, we let p = 1/2 + ε and act as if ε² = 0, it turns out that this is independent of ε and is 93/16, or 5.8125. (Of course, it's not actually independent of ε; for those who know what this means, if we expand the expected number of games as a power series in ε, there's no ε term but there is an ε² term. (It turns out, though, that more World Series have gone the full seven games than would be expected, both because of which games are played at whose stadium and for reasons of baseball strategy.)

So in the "standard" system we have to play 5.8125 games, on average, to get an amplification of 35/16. Could we do better? Let's consider a playoff system that works as follows: the two teams play two best-of-three series. If the same team has won both series, they are declared the champions; if they've each won one series, then a third series is played to determine the champion. This is a best-of-three series of best-of-three series. The amplification is easy to find. If the Phillies have probability 1/2 + ε of winning each game, then we know from the previous post that they have probability 1/2 + (3/2)ε of winning a best-of-three series; thus they have probability 1/2 + (3/2)²ε of winning in this format. The amplification is a bit better than in a best-of-seven series, but only barely. It turns out, though, that on average you play more games in this format than in the best-of-seven format. On average, a best-of-three series consists of 2.5 games. (Half the time, the same team wins both games and it's over; the other half of the time, they split the first two games and the third has to be played.) So a best-of-three series of best-of-three series requires, on average, (2.5)², or 6.25, games. It could take as few as four or as many as nine. (For evenly matched teams, though, the chances of going nine games are one-sixteenth -- a lot less than the probability of a best-of-seven series going its maximum length.)

At this point I started to wonder -- is it possible to design a system that gets substantially better amplification than something on the order of the square root of the number of games played? As I said before, it seems unlikely -- the whole situation reminds me of opinion polling, and if there were a way to do better than the square root there, I'd bet some pollster would have figured it out. Jordan Ellenberg suggested a solution in 2004, in which the World Series would end when one team is up 3-0, 4-1, 4-2, 5-3, or 5-4. I don't like this system for a best-of-seven series, because I was living in Boston when the 2004 American League Championship Series happened -- the Yankees (the real New York team, not the fictional Qankees) won the first three games but the Red Sox came back to win the last four. That could never have happened under this system, although it does open up possibilities for some other exciting series where a team repeatedly just barely avoids elimination and comes back to win.

How long is the average series now? The series can go three, five, six, eight, or nine games. The Phillies win in three with probability p³; the Qankees win in three with probability q³. For the Phillies to win in five, they have to lose the first, second, or third game and win the other four, with probability 3p⁴q; similarly, the Qankees win in five with probability 3q⁴p. For the Phillies to win in six, they must win the sixth game, and three of the first five games; there are ten ways to pick the games they win. But if they win the first three games, then the series would be over, so there are actually nine ways. Thus the Phillies win in six with probability 9p⁴q²; the Qankees win in six with probability 9q⁴p². For the Phillies to win in eight, they must win the final two games, and three of the first six; but if the win the first three or lose the first three, it's over. There are thus 20-2 = 18 ways to pick the games they win, and their probability of winning in eight is 18p⁵q³; similarly, the Yankees win in eight with probability 18q⁵p³. Finally, we use a bit of trickery to determine the probability that the Phillies win in nine. This is some constant times p⁵q⁴. But if we let p=1/2, then we know the Phillies have probability 1/2 of winning the whole series; using this tells us the constant is 36.

The length of the average series turns out to be, then,
3(p³ + q³) + 5(3p⁴q< + 3q⁴p) + 6(9p⁴q² + 9q⁴p²) + 8(18p⁵q³ + 18q⁵p³) + 9(36p⁵q⁴ + 36q⁵p⁴)
which, when p = 1/2, is 369/64, or 5.765625 -- very close to the length of the average World Series in our world. The Phillies' win probability is

p³ + 3p⁴q + 9p⁴q² + 18p⁵q³ + 36p⁵q⁴

which, letting p = 1/2+ε where ε² = 0, turns out to be 1/2 + (147/64)ε. 147/64 is about 2.30. It seems that in a series where about six games are played on average, the amplification will always be a bit greater than two.

But the purpose of the baseball playoffs isn't necessarily to find the objectively "best" team. In fact, a lot of people will argue that a good playoff team is different from a good regular-season team, because there are more off days in the playoffs than in the regular season. This means that once a team makes it to the playoffs they only need three starting pitchers, as opposed to the five they'd need during the regular season. The playoffs don't reward depth. If we wanted to find the objectively "best" team, we'd just award the championship to the team with the best regular-season record. The purpose of the playoffs is to provide entertainment. Since it seems like any playoff format with a "reasonable" average length is about equally good at finding the best team, it seems to me that baseball should stick with its current playoff system because it is familiar, and familiarity enables people to remember how things were in the past and compare those results to the present. Baseball is, of course, a game that is very aware of its own history.

Mark Dominus at the Universe of Discourse makes an argument that homosexuality could be hereditary and yet still not ruled out by natural selection. Basically, the argument is that human sexuality is very complicated and isn't shaped by a single gene (which is patently obvious). We make the assumption that people having more than some number of "gay genes" turn out homosexual and people having less than that number turn out heterosexual. Then in a family where there are lots of people with lots of "gay genes", occasionally one of the kids turns out to be gay and doesn't reproduce, but then this person takes care of their nephews and nieces.

I'm not sure if I believe this, mainly because I know of no evidence that gay people actually pay more attention to their relatives who are not their children that straight people do. (Of course, just because I haven't heard it doesn't mean it's not there.)

Furthermore, on average people share half their genes with their children and one-quarter of them with a niece or nephew. So in order for this to work out in some sort of "expected value" framework, a gay person would have to be able to enhance the survival probability (or, more accurately, the expected number of children, or grandchildren) of their nephews and nieces by twice as much as they'd help their children, if they had them.

However, this could have an effect in times when "expected value" isn't what really matters -- when a family (and therefore a set of people with similar genes) are just barely clinging to life, very close to dying out. The logic then is that a straight person and their gay siblings can put all their eggs in one basket -- and then watch that basket very carefully.

Although I've never heard this sort of argument applied to homosexuality, I have heard it applied to various mental illnesses. There are people who believe that although, say, schizophrenia is obviously very harmful to the people who suffer from it, certain good qualities (say, high intelligence -- I don't remember if this is actually one of them!) tend to occur in the near relatives of schizophrenics. (Let me just say that I in no way am attempting to compare homosexuality and schizophrenia.)

Let's say, hypothetically, that there are ten genes, each of which occur in two variants called "red" and "blue", which cause schizophrenia. Let's say that each of these is "red" with probability p and "blue" with probability q = 1-p. Furthermore, a person which has zero or one of the "red" genes is "normal"; one who has two is of high intelligence, or "smart"; one who has three of more is schizophrenic. Assuming people mate at random, we are in a state of Hardy-Weinberg equilibrium. We compute:

• The probability of a person having zero or one "red" genes is P₁ = q¹⁰ + 10q⁹p.
• The probability of a person having two "red" genes is P₂ = 45q⁸p².
• The probability of a person having three or more "red" genes is P₃ = 1 - (P₁ + P₂). (It can be written out as the sum of the probabilities of having 3, 4, ..., 10 red genes, but it's easier to compute this way.)

Now, consider various proportions of "red" genes in the gene pool; what are the probabilities of a randomly selected person being smart? Schizophrenic?

What we see here is clear. When the frequency of red genes is low, most people are normal. When the frequency is moderate, we see a large minority of smart people and a small minority of schizophrenics. When the frequency is high, the schizophrenics begin to outnumber everybody else. Presumably, then, evolution would want (and here I commit the common sin of anthropomorphizing evolution) a moderate frequency of the "red" genes. As for how that is created, I think that assuming that high intelligence has survival value will do it; when the red genes are rare, "normal" people are the most common but the smart people will out-reproduce them, increasing the frequency of red genes; and when the red genes are common, schizophrenics are the most common but the smart people will out-reproduce them, decreasing the frequency of red genes. But even at the equilibrium point, not everyone will have exactly two red genes, which is what you need to be smart in this model. So there will still be variation.

Something similar actually goes on with the inheritance of sickle-cell anemia; having two copies of a certain allele gives people sickle-cell anemia, but having one copy of that allele confers resistance to malaria.

I live in Philadelphia, home of Comcast. I can see Comcast's new tower in downtown Philly -- when completed, it'll be the "tallest buildling between New York and Chicago" -- from my living room window. An exercise for the reader: where do I get my Internet access from?

If you guessed Comcast, you're right.

Now, I recently had a bizarre customer service with them. The story goes as follows:

Saturday, noon. I'm sitting in my apartment. The Internet and the cable TV stop working, at essentially the same time. I call tech support, wait on hold, and so on. Now, whenever I talk to someone at Comcast's cable television division, they insist "oh, that's an Internet problem, that's not our problem." Hello? They come over the same wire. They go out at the same time. You expect me to believe this is two independent problems/ In the two years I've lived in the territory of Comcast or its predecessors, twice I've lost cable TV and Internet at the same time. I've never lost one but not the other.

Let's say there are three kinds of problems that can happen - one that makes just the TV go out, one that makes just the Internet go out, and one that makes both go out simultaneously. These are all "rare events" -- let's say each happens once a year -- and furthermore I'll say they're Poisson processes. (This basically means that future outages are not aware of past outages, which is technically called "memorylessness" or "independent increments". Now, let's say my cable and my Internet both went out within the last five minutes. What's the probability the same problem caused both?

The probability of losing my cable and Internet due to the same problem in any five-minute interval is five divided by the number of minutes in a year. Since I like RENT (the musical), I know that a year is 525,600 minutes, so this probability is one in 105,120; call it one in 100,000, since everything here is obscenely approximate anyway. But the probability that I lost them due to separate problems? It's the square of this, one in ten billion. So if I lived in this apartment for a hundred thousand years -- ten billion five-minute perioods -- then one hundred thousand times I will lose both TV and Internet due to the same problem. And one time, they'll just happen to go out within five minutes of each other, independently.

Conclusion? Even though I'm complaining about the Internet, it's your job to fix it. But you say "tell your landlord", because of course it's not your company's fault.

Saturday, 1pm. I call my landlord (a small, local property management company; their offices are closer than the closest mailbox, so I walk the rent over there instead of sticking it in the mail when it's due once a month.) The landlord says "it's not my problem", which is what I expected. then I get the tip which will haunt me for the next many days -- "I just let someone from Comcast into the basement of your building."

I run downstairs, chase down a guy in a Comcast pickup truck I see across the street. Someone -- him? one of his colleagues? I'm not sure -- had been there disconnecting people's cable because they weren't paying. I was paid up, but they'd made a mistake. The probabilistic moral here? This will happen less often if you live in a small building. (I've heard this about theft, as well -- the smaller the number of people who have keys to your building, the less likely your property is to be stolen.)

Another moral here is that Comcast ought to have a better system for telling which wire services which apartment, but that'll come later, when I talk about how their current system seems to work.

As I wondered earlier: there are prediction markets in which you can bet on Harry Potter living. NewsFutures puts his probability of living at [redacted; click on the link if you want to know]. You can see a chart going back to March here. It's currently half past six in England; the answer will be publicly available in five and a half hours. Interestingly, the contract pays 100ドル (of virtual money) if Harry lives, 0ドル if Harry doesn't live, and 50ドル in "any other case".

Also, Amazon.com reveals that the Harry-est town in America is Falls Church, Virginia, as measured by the largest number of pre-orders per capita. (Only towns with more than 5,000 people were included.) The top twenty-two cities on this list are all within fifty-two miles of a major city. (Why 52? I was saying 50 at first, but Fredericksburg, VA was just outside the cutoff.) They are:

• Falls Church, VA (Washington, 10 miles)
• Gig Harbor, WA (Seattle 44)
• Fairfax, VA (Washington 21)
• Vienna, VA (Washington 16)
• Katy, TX (Houston 29)
• Media, PA (Philadelphia 22)
• Issaquah, WA (Seattle 17)
• Snohomish, WA (Seattle 31)
• Doylestown, PA (Philadelphia 40)
• Fairport, NY (Rochester 11)
• Woodinville, WA (Seattle 20)
• Princeton, NJ (Philadelphia 45; New York 51)
• Webster, NY (Rochester 15)
• West Chester, PA (Philadelphia 37)
• Williamsville, NY (Buffalo 11)
• Fredericksburg, VA (Washington 52)
• Port Orchard, WA (Seattle 22)
• Decatur, GA (Atlanta 6)
• Larchmont, NY (New York 27)
• Downingtown, PA (Philadelphia 39)
• Canton, GA (Atlanta 41)
• Woodstock, GA (Atlanta 31)

What surprised me was that these twenty-two cities are suburbs of a small number of cities; in particular Harry Potter seems to be most popular in the suburbs of Washington, Seattle, Philadelphia, and Atlanta. I wonder if this would correlate with the fact that certain metropolitan areas are full of the type of people who order things on the Internet than others; Seattle has that reputation, at least in my mind, and they're also Amazon's home. But so do the Bay Area and Boston and those aren't represented there. (Sebastopol, sixty miles north of San Francisco, is #30; Mill Valley, fourteen miles north, is #36. Not a single town in all of New England is on the list.) I don't have population figures at hand but I would guess that the cities on this list are smaller than the average city in the sample. You'd expect to see larger fluctuations from the "average" in smaller cities. And it wouldn't surprise me to learn that these particular towns are underserved in terms of having bookstores nearb,y, compared to how many readers there are in the city, thus making it more convenient to order online. Perhaps the bookstore chains should look at this list to decide where their next locations should be?

The Harry-est states in America is a different story; you'd expect from the first list that the states with lots of suburban population -- New Jersey or Virginia comes to mind, both states with no really large city within their borders but with one just outside -- would appear high up? Perhaps -- but the winner is actually the District of Columbia. (As a city, however, D.C. doesn't even make the top 100.) The six New England states are all high up -- Vermont is the highest at #2, Rhode Island the lowest at #16.

The moral of the story is that depending on how you sample you get very different results. Of course, the whole "Harry-est cities/states in America" thing is just a silly Amazon promotion.

Geometry Saved Me Money, from Binary Dollar, via Grey Matters. Which is more: a twelve-inch pizza or two eight-inch pizzas?

The twelve-inch pizza, of course; it is more square inches of pizza. (I'm assuming all pizzas are equally thick.)

However, if you really like crust, the two eight-inch pizzas might actually be the better deal. One twelve-inch pizza contains 36π square inches of interior and 12π inches of crust; two eight-inch pizzas contain 32π square inches of interior and 16π inches of crust. So if you're willing to trade 4π square inches of interior for 4π inches of crust, take the smaller pizzas. That is, if you'd rather have a third of the crust of the pizza than a ninth of the interior, or if you'd prefer three crusts to one crustless slice.

I like the crust, so I might.

(This analysis assumes, of course, that the thickness of the crust is negligible, so that "an inch of crust" actually means something.)

The waitress in the restaurant where this question came up thought the customers would prefer the two eight-inch pizzas because it was more "slices of pizza". Maybe it's just me, but a "slice of pizza" is a meaningless unit, because it's not standard. I would have at least expected the argument that eight plus eight is more than twelve.

I suspect that pizza is the food that is most often "illogically" priced. I've seen, say, chicken wings sold at "10 for 5,ドル 15 for 8ドル" but you don't see that too often, because most people can do the math and realize that buying 15 is a bad deal. (Think of it this way: what if I want thirty? I can get three 10-packs for 15,ドル or two 15-packs for 16ドル.) But with pizza people will throw up their hands. Also, I have seen places where a larger size of pizza costs more per square inch than a smaller pizza (I can't find any right now); I was once told that this was because for whatever reason the large size was more inconvenient to make (it fit in the oven funny, for example). That at least seems like a plausible economic reason; it's clearly not the cost of the ingredients and almost certainly not the labor.

An Associated Press article claims that increased temperatures cost consumers money at the gas pump. (the link is to MSN Money Central.) The reason is not the usual seasonal fluctuations in gas prices, but rather the fact that gas is priced by volume, but the chemical reactions that fuel a car don't care about the volume of fuel you put in, but rather the mass. A United States liquid gallon has historically been defined as 231 cubic inches (although I suspect that now it, like most other units in our system, is defined as some exact multiple of some metric unit, in this case the cubic meter, liter, or cubic centimeter). But since gasoline, like other materials, expands as the temperature goes up. The article says that gas pumps "always dispense fuel as if it's 60 degrees". I assume this means that when it's warmer than 60 degrees, the "gallon" of gas you get from the pump is still one gallon by volume but it's slightly less mass -- about one percent less at 80 degrees than at 60 -- than it would be at 60 degrees and therefore your car doesn't go as far.

Consumer advocates are up in arms about this.

Personally, I'm a bit suspicious. For one thing, gas prices fluctuate wildly. If you are not blind, you can see this. (I mean this literally -- anyone in this country with eyes has some idea what gas costs, even if they don't buy any. I personally don't buy any -- I don't drive -- but I know that regular gas costs 2ドル.96 a gallon right outside my apartment. Then again, I live across the street from a gas station.) I have no doubt that gas station owners are aware of this effect. Their margins on gasoline, depending on who you believe, are anywhere from 2% to 10%; this effect could seriously eat into their margins if they didn't know about this. The real money in running a gas station is from the attached convenience store.

I suspect that the news media over-reports on gas price fluctuations, though. "Gas prices went up this week" or "gas prices went down this week" gets people's attention; "gas prices stay the same this week" doesn't.

Second, a gallon is a unit of volume. In fact, I'm not totally sure what this calibration to 60 degrees even means. The case would seem a lot less silly to me if gas were being priced by the kilogram. (This leads me to wonder -- if gas were priced by the pound, would people complain that at higher altitudes gravity is weaker so a pound of gas has greater mass?)

Third, fitting the pumps to correct for temperature would be expensive -- a few thousand dollars a pump.

Fourth, 60 degrees was probably chosen because it's pretty close to the average annual temperature in much of the United States. So this might help people in the summer -- but it'll hurt them in the winter. (Of course, it is easy for me to say this, as both a non-driver and someone from a place where the average annual temperature is 54.3 degrees Fahrenheit.)

Still, it wouldn't surprise me if there are some people avoiding buying gas on hot days so that they can save a few cents a gallon. I'm not sure how viable such a strategy is, in part because the gas is stored in underground holding tanks and underground temperatures don't vary nearly as quickly as air temperatures. Second, what kind of savings are we talking here, a couple cents a gallon? Is it really worth stressing out over this to save thirty or forty cents on a tank of gas?