Air Travel Safety

I found this website, basically its a list of all plane crashes in the US going back 15 years, its kind of insane how safe air travel is.

Looking at serious plane crashes (planes with a capacity of 10 or more with at least one fatality), there have been only 14 since the year 2000. The fatality count by incident is below:

3
44
49
20
13
21
260
92*
65*
64*
45*
82
109
19

*includes the 9/11 hijackers in fatality totals.

The total number of people killed in while on major incidents is 886. 867 if you don’t count the 9/11 hijackers.

The vast majority of them happened in four instances: a 747 from Taipei to LA in October 2000, the Concorde crash in July 2000, the bizarre crash of an American Airlines plane in New York city just months after 9/11, and the September 11 plane crashes themselves.

All of those took place in 2000 or 2001; since 2002 there have been no crashes with 100 or more fatalities in the US (including flights into our out of the US).

Since 2000, there have been about 10 billion passenger flights in the US (about 2 flights per person per year), or about 1 in 11 million chance of dying on any given flight.

If we only look at the last ten years, we get an even better picture, 116 fatalities with 7.2 billion passengers, for a one in a 62 million chance of dying on any given flight. The past five years it becomes one in a billion chance; although with only one incident it probably understates the chances somewhat.

What’s equally crazy is the change in safety numbers. In the 1990’s a US passenger had about a one in 5 million chance of dying in any given flight, which is pretty damn safe; in comparison it was about as dangerous in the 1990s to board any given flight as it was in 2007 to drive 16.5 miles. The 2000s have been about 70% safer (or if you prefer, 42% less dangerous), meaning that boarding any given flight is about as dangerous as driving 9.5 miles (in 2007 miles). The past ten years? Well, you’re about as likely to die from boarding any given US flight as you are from driving 1.15 miles. And if we limit our data to the past five years, boarding any given flight is as dangerous as driving the length of a football field.

Of course, the last five years probably aren’t actually a good indicator; if we’re in a point where air crashes happen on average once every 10 years, then looking back five years will necessarily give you a bad estimate, (ie, either higher than average, if the previous 5 years had a plane crash, or lower than average if it didn’t). So you’re chances probably aren’t one in a billion of dying when boarding a flight, they’re probably more like one in a hundred million or so.

All this is to say that, as measured by safety, the US has done an incredible job at promoting flight safety. Regulators have a clear mandate to make things safer, there isn’t much of a opposition group (while there are people who might want fewer regulations in principle and some people who might want to cut corners on safety, nobody is against aviation safety), and can be clearly measured. When these things happen, well, you get government success; the NTSB and FAA are examples of government greatness. Boring greatness mind you, and perhaps they are impressive because they’re boring. We have been able to take something mad, to travel at a speed of 500 miles per hour suspended miles above the ground by nothing but air, and have made accidents as unlikely as powerball victories.

Sources:

Number of Air travel passengers:

http://data.worldbank.org/indicator/IS.AIR.PSGR?page=3

Number of auto fatalities and air fatalities before 2000:

http://www.rita.dot.gov/bts/sites/rita.dot.gov.bts/files/publications/national_transportation_statistics/html/table_02_01.html_mfd

Number of air fatalities since 2000:

http://www.airsafe.com/events/us_ten.htm

Snowflakes

You are one in a million. A precious snowflake, and you’re totally unique.

What was your reaction to those statements? My guess is that you may think it is hackneyed, untrue and ridiculous. You aren’t unique, there are tons and tons of people just like you. Lets find out exactly how many.

First, lets define a few characteristics here.

1: Gender
2: Race
3: Age Group

For gender, there are two possibilities, for race, lets say 3 (White, Black, Asian) (you could argue a lot about the appropriate number), for age group, lets say six groups (preteen, teen, young adult, adult, middle age, elderly). We could expand further downward (infant, toddler, etc), but you aren’t one of those because you can read.

So as of right now, there are 36 different possible readers taking only 3 characteristics (which of course aren’t the full dimension).

Lets keep adding some values:

4: Political affiliation (3 possibilities liberal, conservative, libertarian)
5: Religion (we’ll go with 6 common possibilities, Christian, Muslim, Jewish, Hindu, Buddhist, Atheist)
6: Nationality (google say 196, but we can differ)
7: Occupational sector (I could argue a lot about these and it depends on how you define these, but lets say 20 different employment sectors)

So now, based on these 4 new characteristics, we have 3*6*196*20, or 70,560, times 36 (from our first three), which gives up 2,540,160.

But we’re not anywhere near done. For fun, lets add the Myers Brigs Categories

8-11 Myers Brigs (Extroverted/Introverted, thinking/feeling, judging/perceiving, sensing/intuiting, or 16 total)

Lets add some more random items:

12: Are you a traditionalist or a trailblazer? (2)
13: Are you high strung or chilled out. (2)
14: Are you career oriented, family oriented, friend oriented, or community oriented (4)
15: Do you get your way or get along? (2)

Ok, lets go back to some more traditional measures:

16: Sexual Orientation (2)
17: Education Level (we’ll say 5, did not finish high school, high school only, some college, college degree, graduate degree)

Ok, now we’ve got 16x2x2x4x2x2x5 which is 5,120. Multiply this by our number above and we get 13,005,619,200. With only 17 criteria, we now define more types of people than there are people in the whole world. So if you’re a young adult white male conservative Christian American financial services professional who is an INTJ, a traditionalist, a high strung community oriented get-along straight college graduate, you can say that you are unique. Actually you can’t, because that’s who I am.

Now, there’s two major complaints I anticipate. The first is that my categories above weren’t fair because your religion/race/whatever wasn’t on there. Well, that just means you are even more unique than above, you truly are special.

The second, and much more relevant piece, is that many of the above are correlated with one another. Take me, for instance, a straight white christian male. All of those are somewhat to heavily correlated with being conservative, which, no surprise, I am (kind of). Which means that, although there are in fact more categories than there are people to fill them, people aren’t randomly assigned, and its very likely that there a bunch of categories with more than one person in them, some of them probably numerous (perhaps mine has a quite a few). But it isn’t many; lets go by my example again.

First, lets start off with Americans, which we’ll say there are 300 million of.
Half are men, which gets down to 150 million.
My age group has 20.3% of the population (I’m defining young adult as 20 – 34), that’s 30.45 million people.
We’ll save 78.5% are christian, of which half are conservative (total of 39.25%). Brings us to 11.95 million.

4% of people work in financial services, and we’ll say that all of them have at least an associates degree. According to wikipedia, 41.89 % of the population had a associates or bachelors degree and 11.77% had advanced degrees. We’ll assume that the ratio of advanced to non advanced degree holders is the same in the financial services industry as it is outside of it, which means that 2.88% of people in the US are financial services professionals without an advanced degree. This now brings us to 344,160 people are like me, in the first 6 categories.

According to google, INTJs are 2% of the population, but more common in men, (of which I am), and we’ll say that in other various ways I’m more correlated with the value, so to pick a number we’ll say 8% of people with the above characteristics are INTJs.

Of these, lets say 80% are traditionalist, 70% are high strung, 40% are community oriented, 75% get along and they’re all straight.

Which means that, in my bucket, there are perhaps 4,625 people. Which may be a lot, but its not too many. I’m sure there are bigger buckets, something with Indian Hindu working in agriculture probably has many more, but again, we’re only talking about 17 different characteristics. I haven’t mentioned whether you like history, whether you are a poet, if you play an instrument, if you prefer the country to the city, the mountains to the beach, the summer to the winter, whether you run, bike, play a team sport. If you like science fiction, if you watch reality TV, if you are in to fashion, or follow sports. If you do drugs, drink alcohol. If you’re in a weird subculture.

All of these are important things, things which matter, which help describe who a person is. And I haven’t even mentioned whether you’re married or have kids yet.

Combine all of these, and while I suppose its still possible there are “Buckets” which might have more than one person in them, I imagine that, just within these characteristics, the vast majority of humanity has a bucket entirely to themselves. And I can go on regarding these characteristics, I’m sure I can name another 20 or 30 categories easily, and there are numerous subcategories within these groupings.

So what is the point? First of all, yes, you truly are unique, there is only one you, and we’ve just proved it, with math.

Sometimes, we compare things to other things. Just the other day, I contrasted the US political climate to the number of original movies coming out. That was a comparison I made, because of my outlook and experience. There are comparisons, accomplishments and insights that only you can make. You are special, and I mean that in the best possible way.

Are ghosts real?

We all have guilty pleasures, mine is paranormal activity. Not the movie series (I’ve never seen any but have heard they are horrible), but just the concept that bigfoot or ghosts or aliens are real.

Three reasons bigfoot isn’t real:

1: If he were real we would have hard evidence: remains, a captured creature, or unmistakable photographic evidence. We have little “good” photographic evidence.

2: Sightings correlate with interest over time. We’ve had few sightings before 1958, suddenly he appears not only in the Pacific Northwest but across the country, as far east as Maine and as far south as Florida. Basically, we’re to believe that he hasn’t appeared in the US at all, until he becomes culturally important, then he is suddenly ubiquitous.

3: The more likely one is to believe in Bigfoot, the more likely one is to find evidence of him. The most famous piece of evidence is the Patterson Gilmlin film (you almost certainly know what it is, but if you need a refresher, click here). Why were the people in the area? They were filming a fictional movie about bigfoot. There have been who knows how many movies filmed in California, thousands of people in the Californian mountains and woods, and the single best piece of evidence happens to be created from people already connected to bigfoot? It seems like a very unlikely coincidence.

There is one reason, however, to put some stock in the bigfoot legend. Take a look at this map:

Bigfoot_Sightings_in_USA

(map by wikipedia User:Fiziker)

Bigfoot sightings seem to follow a general rule: the more people in a state, the more people who see Bigfoot. (New York doesn’t follow this rule, but this makes sense, most people in New York are in a single very urban metropolitan area). Almost everything on this map can be explained by a simple rule: when people are in rural areas, they see a bigfoot with some constant probability. That probability should depend greatly on not just the number of people in an area, but in the number of bigfoot as well, and it doesn’t seem to. There are few to no patterns with regard to geography within this map.

Except one. Washington has more than any other state, and Oregon has a disproportional number of sightings. Why is this? Is it some cultural reason, are people expected to see bigfoot more in those states? Are people more likely to perpetuate hoaxes in Washington and Oregon?

I think that either of those explanations is a better one than that Bigfoot actually exists, but I must admit that it does influence my belief somewhat (perhaps from a 0.2% to a 0.3% chance that bigfoot exists), there does seem to be some geographic pattern that bigfoot follows.

Regardless of all this, bigfoot isn’t a big deal. If somebody were to discover a bigfoot next year, it’d be a big story, then life would go on as normal. If bigfoot were to exist, it wouldn’t really affect our lives or our concept of our world at all. There are other such “cryptids” which are similar, the yeti, the skunk fox, ogopogo, etc. I doubt any of them exist, but they wouldn’t be a big deal.

Ghosts are another matter. If we ever determine that ghosts actually exist, many people will have to re-examine their entire worldview. Suddenly, physics will either have to deal with the fact that we have souls or that there is something really weird going on with ghosts.

If you were to ask anyone why they don’t believe in ghosts, the answer would probably be something like why hasn’t anyone seen them. But thousands, perhaps millions of people have seen ghosts. Well, why aren’t there any photographs of ghosts? But there are hundreds of photographs of ghosts. Don’t believe me, do a quick google search for ghost photograph, and see them.

But are they real? Is it possible that every single person who attributes something to ghosts is wrong about it? Yes, it is possible. Is it possible that every single photograph and video which has a ghost is some form of a hoax, optical illusion, or other mistake? Again, it is possible, people make mistakes about things all the time. We see faces all the time where they shouldn’t be.

So should we believe in ghosts? Quite frankly, I have idea. The amount of ghostly evidence seems to about fit the pattern of people frequently mis-attributing things to ghosts and sometimes making things up. But I have no idea what the evidence level of “ghosts are real” would look like.

Lets put ghosts in perspective by talking about bigfoot. If bigfoot was real, there’d probably need to be at least 2,000 of them to form a stable breeding population. If the population was spread out over the state of Washington, (area of about 70,000 square miles), and we were to monitor 5,000 sites throughout Washington (either through game photographs or people with cellphone cameras or the like), each of which covers a half acre. Finally, each bigfoot stays in a viewing site for 6 hours, then is randomly transported to another viewing site. This is an admittedly weird scenario, but it gives us something to go by; some assumptions are simplifications to be sure (bigfoot probably aren’t teleporting), but they go in both directions (a half acre may be too big, but six hours may be way too long.  Also, there are areas of Washington we shouldn’t expect Bigfoot to live, such as Seattle).  Anyway, lets see what happens when we run the numbers:

This boils down pretty quickly to a randomly drawn ball problem. If there are 91 million balls in an urn and 2,000 of them are red, the chances of drawing a red ball are only .00219%. Which means that there is a 99.99781% chance that, at any given time, you will draw a non red ball. The chances that you draw a red ball in either of two draws is the probability that you don’t draw a red ball in the first times the probability you don’t draw one in the second. To simplify, we can reduce the probability of drawing a red ball on any number of draws by using the formula x = 1- P^N, where x is the probability of not drawing a red ball in N draws, and P is the probability if not drawing a red ball in one draw.

We can use our example and ask a new question, how many draws would we need to make before we had a 50% chance of drawing a red ball? We can use a simple formula, .5 = .9999781^N. Or, to make it simpler,N = Log(base .9999781) of .5, which means N is 31,656.85. So to get better than even odds of drawing a red ball, we would on average draw 31 thousand balls.

Lets apply this to bigfoot. If there are 2,000 bigfeet in Washington, and we have a viewing sites of a half acre each, that means we would have to make 31 thousand observations before we expect to see one. Which, with 5,000 viewing sites, each observation taking 6 hours, would take us all of 36 hours. A day and a half to find bigfoot, given our parameters. Which means we should be seeing many many many more bigfoot sightings than we do right now; about 120 per year, not 593 all time. Now you can argue about my numbers, but that’s not the point. The point is we can have some sort of baseline, we can make assumptions to create some sort of a test, if bigfoot is real, how often should we see him. We can then get an estimate and compare it what we actually see, and see if it matches our theory.

I don’t think we can do that with ghosts. If ghosts are real, how common are they? I have no idea. If they exist, what are the chances that we see them in any given time period? Again, I have no idea, it matters a lot if they are supposed to come out once per day or once per year. Are they permanent or do they eventually dissipate (or cross over or whatever). I have no idea what the answer to any of these questions would be.

I’m not trying to argue that ghosts somehow exist – I am saying that i don’t even know how to determine if they are.

Perhaps the wisest thing I’ve ever heard in terms of epistemology is this: that true things become more obvious the more they are studied. I think that’s the only thing I can really say about ghosts, that they’ve been studied a lot, but we aren’t more sure of them now than we were 20 years ago. Perhaps that’s something, but its hollow to me, a non answer. Sure, I can say that I don’t believe that ghosts definitely do exist, but I have a very hard time understanding why I should have a strong belief that ghosts don’t exist.

The great filter Part 2

A while ago I wrote about the “Great Filter,” or the reason why we don’t see aliens everywhere we look in the universe. Read about it here:

Last time, I argued that the great filter cannot be a totalitarian regime, is very unlikely to be either berserkers or environmental damage, and is somewhat unlikely to be nuclear war and/or pandemic. Which leaves us with two more filters, the starships are hard, or that civilizations aren’t interested in colonization.

Today, I’ll talk about whether starships are hard.

In order for something to be a filter, it needs to have the following characteristics.

1: It must prevent the colonization of the galaxy.

2: It needs to be stable (or long-lasting), it if effects a civilization in time period x, it must still do so in period x+1

3: It needs to be universal, and effect (nearly) all civilizations, regardless of biology or culture.

So does difficulties building starships meet all three categories? For item 1, definitely. For item 2, also definitely, the laws of physics governing space travel aren’t changing. If its hard to build a spaceship today, just waiting won’t make it easier. At first glance, it appears to be universal as well, as all civilizations are facing the same laws of phyics. However, there may be two reasons why this would be different. First, some species may be more able to survive on starships, for instance they may be smaller. Secondly, some civilizations may start out closer to colonizable star systems than others. However, even with these two, we can say that space ships hit the “nearly” universal tag.

So, having established that difficulty in building spaceships can lead to Fermi Paradox, we tackle the more interesting question which is, has it?

At first it seems obvious that it will. Assume that the highest speed of a starship is 10% the speed of light. Next, assume that the average colonizable target is 80 light years away. Simple math says that it will take 800 years, or roughly 23 human generations to complete the journey. So you have to have a starship big enough to house enough people to preserve genetic diversity over such a time period (we’re talking about hundreds at the bare minimum, probably more realistically in the thousands), and enough space to grow food, house recycling functions (for not just materials but water, air, etc), provide living quarters, and enough energy to run the whole thing. Furthermore, we’d need to transport all sorts of animals, fish, livestock etc, to populate the new world, in addition to feeding people along the way. Also, all the equipment needed to actually colonize the new world. We’re talking about a big spaceship, and enough energy to get that spaceship that fast is enormous, not to mention slowing it down when it gets to the target.

All this seems to lead us to the conclusion that yes, space colonization is very hard.

But there are a few things we can do to modify this. First, let’s assume that people aren’t busy living on board the spaceship, but instead in a state of suspended animation. Ditto for all the cattle, fish, plants, dogs, and whatever else we want to bring. Suddenly, the total power requirements goes down, a lot.  Also, since we don’t have to worry about storing or growing food, we can cut our speed down, say to about 2% the speed of light, making the journey take 4,000 years instead of 800, which means that civilization on earth may no longer exist, but if the new colony can become self sufficient and expand, then it will be able to colonize the galaxy.

One might object by saying that I’m making up a technology that we haven’t proven to exist. Furthermore, while suspended animation may be possible, it might not work for long periods of time (it may work for 50 years, but not for 4,000), or that it may still take a lot of energy to keep the suspendee alive. All of this may be true, but I would argue that of course we assume there must be some technology which doesn’t exist for us yet which could lead to space colonization after all I don’t think we’ve discovered everything. And while there may be difficulties in suspended animation, there is nothing that I know of in the laws of physics which would prevent it, unlike warp drives for instance.

There is however, a much easier way to transport people across the vastness of space than suspended animation. And that is to transport not fully grown humans, but fertilized eggs. While we certainly don’t have experience freezing embryos for thousands of years, there’s no reason to assume that storing them at near absolute zero temperatures wouldn’t work. Furthermore, in the coldness of space (2.7 Kelvin), you wouldn’t need to spend energy on refrigeration technology. Now, we’d need a way to take those embryos and develop them outside of the womb, and then raise/educate those children, which could either be done by a subset of humans (if suspended animation is feasible), or by robots.

The same holds true not just for humans, but for all manners of plant and animal life, we can take an entire genetic ecosystem worth of genetic material in a series of canisters no bigger than a large room. And there may be even more compact ways. Instead of storing embryos, we could potentially just store DNA sequences of organism, then “build” them when the starship reaches its destination. Whether this is feasible or not is up in the air, but it certainly seems possible to me.

Now, how big would a spaceship need to be in order to do this. How about 10,000,000 metric tons of starship? That’s big, about 15 times the mass of the largest ship every built (the Seawise Giant), but small compared to something like the fleet of oil tankers on planet earth right now (less than 10% of the mass of Ultra Large Crude Carrier, when loaded with petroleum). Now, can a ship that size hold all the things needed to colonize a planet? Truth is, I have no idea, but lets run with it for a second.

So if we have 10 million tons, that’s 10 billion kilograms. Doing some math (e = 1/2 mv^2), it will take about 3,000 years worth of current us electricity consumption to get the ship up to a speed of 1% the speed of light, which is a lot, but is it too much? If we reach the point of harvesting energy using space solar panels, it becomes a bit easier. We would require only a solar panel of 136 miles on each side, placed at 0.1 AU, to harvest that amount of energy over a year (assuming 22% efficiency). This is about one and a half Marylands worth of solar panels. This seems like a lot, but (according to this source: http://landartgenerator.org/blagi/archives/127) it is only about half the total surface area of highways in the US. In short, if we get to the point where we’re mining asteroids, we can do it. Storing that energy and then using it to power the ship are another matter, and while it seems hard, it doesn’t seem impossible. (Math allows us to proportionally change things easily! If you want to increase the speed of the ship, to .02 c, for instance, just double the length of the solar panel side. If you want to double the mass of the ship, use two years instead of one).

One final thing, which I’ll comment on because I spent a long time figuring this out, is how to slow the ship down. There are plenty of actual starship designs out there, including hyrdogen scoops and the like (Bussard Ramjets), but I thought, hey, why not use a parachute to slow a ship down. I did the math to determine how big a parachute you’d need to slow down such a ship, and it was one of my favorite problems to solve ever. Feel free to skip all the math, but here it is:

To solve it, we use the drag equation, f = 1/2 p * v^2 & a * c

Where F is force, P is fluid density, V is velocity, A is area of the parachute, and C is the drag coefficient. For reasons I won’t go into, we can say that C = 2, so the formula becomes

f = p*v^2*a, or
v’ = -p*v^2*a/m (m is mass)

I put the negative sign in because the acceleration will always be negative, ie the ship will always be slowing down. In order to translate this to a function of time, we use the initial value problem: (http://en.wikipedia.org/wiki/Initial_value_problem)

Since -p*a/m is a constant, lets just call it k, that gives us

dv/dt = v^2 k

next

dv/v^2 = k* dt

Integrate both sides and you get

-1/v + C = k * t + B

subtract C from both sides

-1/v = k*t + B-C

We can call B-C a new term (D), then simply isolate v

-1/v = k*t + D

1/v = -k*t – D
v = 1/(-k*t – D), which means that the function is

v(t) = 1/(-k*t – D), we know everything except D, but we know what the starting speed is (.01 c, or 2,997,925 m/s)

v(0) = 1/(-k*0 – D) or

2,997,925 = 1/-D, which gives us a D of -0.0000003335640952.

so v(t) = 1/(-k*t + 0.0000003335640952)

k, if you remember, is -fluid density * area / mass. So for units of k*t we get (mass / length^3 * length^2 / mass * time), this reduces down to 1/length * time, or -1/velocity, which is great because that’s the unit we need, our units match.

Now, it would be absurd to use this to slow the spaceship down to zero (it would take forever), but we can use it to slow the starship to say the speed at which the earth revolves around the sun (30,000 m/s). Finally, lets say how long we want it to happen (over a period of 3,000 years, for instance), we get

v(3000 years) = 30,000 m/s

Since we never defined how big the parachute is, we can now solve for it, given our constraints above:

v = 1/(-(-P*A/M) *t + D))

Rearrange to isolate A:

A = M/pt * (1/v + D)

throw in the numbers we know: (I originally did the math based on a 824 thousand metric ton ship)

A = 824,000,000 (mass of ship) / 2.39E-21 (density of space in kg/m^3) * 94,672,800,000 (3000 years in seconds) * (1/30,000 (speed in m/s) -0.0000003335640952 (our constant D, in s/m)

I love this so much because it uses ridiculously large and small numbers (giant ships, giant sails, the density of outer space!!)

Anyway, we get a value of 120,123,349,601,661, square meters, which is pretty big, or 10,000,000 meters on one edge of the (square) sail, about 6,850 miles, which is a sail about the diameter of earth. A sail built from any substance would be prohibitive in terms of mass, but using an electromagnetic field wouldn’t.

All of this is to say that I think it would be possible to slow the ship down. Can we speed it up? Put it this way; 1% of the speed of light is only 200 times faster than space probes we’ve already built. Surely we could built something to go, if not that fast, 50 times faster than the Voyager 1 spacecraft.

All this, though, leads us to the easiest path of all, while humans may or may not ever be able to colonize the galaxy, surely self replicating robots could? We’ve already built robots which can function for years on other planets, building some sort of robot or collections of robots which could construct more versions of themselves on other planets makes not only the difficulties of getting there, but the difficulties of transporting humans there almost disappear.

Building a collection of robots to colonize the galaxy might not seem romantic or noble, and it may not even be wise; in fact we might say that it is a very bad idea. But that doesn’t matter for our purposes, all we need is the idea that a: it’s possible and b: that somebody somewhere decides to do it. If we have those two conditions, then its pretty much inevitable that we get a galaxy full of robots, which, based on our observations, doesn’t appear to be what we have.

Starships are hard to build, no question. But I don’t think they are so hard to become the great filter. If there are enough intelligent civilizations, one of them will build self replicating robots and conquer the galaxy.

Next up, do we lose our desire?

Playing the lottery is not a Sharpe choice

The Powerball Jackpot is approaching $500 million dollars tonight.  The chances of winning are 1 in 175 million. Which gives the expected value of $2.86 per ticket, and at a cost of $2 per ticket, that’s a great value!

Of course, the 500 million is an annuity, the “lump sum” or the real value is 337, which gives a value of $1.92 per ticket. This is increased by all the other prizes, as one can win $4, $7, $100, $10,000 and $1,000,000 for various combinations. Doing the math (which I won’t spell out in this post) gives you about $0.36 per ticket, so for tonight’s jackpot we can estimate the expected value of a single ticket at $2.28, or roughly a 10% average return.

If, as we’ve calculated, the expected value of the ticket is $2.28, is buying a ticket the economically intelligent thing to do? What, for that matter, does expected value actually mean?

One way to think of it is to imagine buying all possible tickets. There are 175 million possible number combinations (thus the 1 in 175 million chances), if you bought every possible ticket, you would be guaranteed a winner, (and multiples of all the lesser prizes) which means that buying all the tickets will make you money, about $400 million to be approximate. If buying all the tickets will get you x amount of money, then buying 1 ticket will get you x/175 million in money; doing this math gets us to the $2.28 figure.

But there’s something else to consider, that is the possibility of another winner. If two people win the prize, then the prize amount drops in half. Lets say, for the sake of simplicity, that there is a 40% chance of having two winners (which, if anything seems too low, it seems like when lotteries get this high). That reduces the expected value of the ticket from 2.28 to 1.90, which makes it fall below our magical $2 level.

But what is this telling us? Well, it tells us that if we buy all the tickets, now we will expect to lose money. But why does this have any relevance for anyone who buys a single ticket? Lets imagine a scenario. First, lets assume that, for whatever reason, the jackpot is rarely split in two so you barely figure that into your calculations (maybe a 1% chance of two winners instead of a 40% chance). Now, imagine that you buy a Powerball ticket, and to your joy while watching the drawing you see each of your 6 numbers match those being drawn. As you’ve just one the jackpot, you are ecstatic. The next day, as you get the morning newspaper, you see the headline: “Record Powerball Jackpot has 2 winners” Instead of winning the full jackpot, you only win half. Now, do you regret buying the ticket? Of course not, ex ante, the math worked in your favor, ex post, the math worked in your favor.  So why, if when event x happens it does not cause you to change any rewards scheme, should the possibility of event x enter into your calculations?   No realistic number of winners will ever push the jackpot down to a level where you wouldn’t be happy winning it.

Lets change gears a moment and talk about taxes. Assume that taxes takes away half the winnings.  Business Insider did the math here: basically the taxes push the expected value to $1.32 (for some insane reason they are reporting the expected return on buying a ticket, instead of the expected value of a ticket.) They figure that you should apply the after tax return to the cost of buying a ticket to determine the expected value.

But what does our analysis actually mean? If we return to the scenario of buying every ticket, all in all we’d spend about $350 million, get the jackpot and all the other prizes, for a value of about $400 million. Now, if you assume that you would pay half that $400 million in taxes, you’d be left with $200 million, or a loss of $150 million, so you figure don’t buy the tickets. But in reality you wouldn’t be charged $200 million in taxes, because of the ability to write off gambling losses against gambling wins. So instead of paying $200 million, you would only pay $25 million in taxes (on 50 million net winnings), which pushes the net value to positive territory.  So which value is correct? One the one hand, you’re not actually buying all the tickets, and the value of a single ticket as a tax write off is negligible. On the other hand, it’s not like paying the taxes on the winnings is going to make a big difference, you’ll still be horribly rich. So what is the answer?

The answer is that there is no answer. Using something like expected value is a tool which helps us understand our world, but it is not the world. For any Powerball lottery, there is always a very small chance that you will become very rich. Exactly how rich is hardly an interesting question. Imagine yourself with $100 million. Now imagine yourself with $200 million. There isn’t much of a difference. Or to put it differently, think of all the things that you could do with $200 million that you couldn’t do with $100 million: not a gigantic difference. So if the human difference between 100 and 200 million dollars isn’t big, what is the point in taking the jackpot, multiplying it by the probability of winning, and comparing that number to the ticket price? If you win, you’ll be happy. If you lose, you wasted money. In other games with lower variance in returns (such as roulette) can help you understand whether you will win or lose over the long run, in the case of roulette the long run is over an hour or a few hours. But in Powerball, the long run would last millions of years (potentially billions if you only play when the Powerball odds are in your favor), so how is that useful?

In finance, there is a concept called the Sharpe ratio. It measures the marginal change in return to marginal change in risk. Basically, it’s a way of adjusting returns for risk. If I did my math right, in our situation we an expected return of about .14 (winning 2.28 on a 2 dollar ticket is a 14% return, we can assume the risk free rate of return is zero overnight (close enough anyway), and a standard deviation on the return of 162 million. Which gives us a Sharpe ratio of 0.00001123293428; well below the advised value of 1 for a decent risk adjusted investment. It’s not perfect resolution of the problem, but it does illuminate the basic problem with Powerball, if it’s worth $2 for a 1 in a 175 million chance to win $300 million, it’s probably still worth $2 for a 1 in a 175 million chance to win $40 million, big changes in very low probability events just aren’t significant.

Superbowl Squares

As yet another service in my not a football blog, I’m going to talk about gambling a little bit, specifically the idea known as superbowl squares. What happens is, you have a ten by ten grid, “buy” a square within that grid for a certain amount of money (usually 5 or 10 dollars, however I’ve heard rumors of $100 per square or higher), then the each row and column has a digit between 0 and 9 randomly assigned to it, the rows represent one team and the columns another. Thus, each square has a value, for instance Patriots 2, Seahawks 9.

If, at the end of a quarter, the Patriots score ends in a 2 and the Seahawks score ends in a 9, that square wins, usually a quarter of the whole prize (so for $5 a square, the winner would get 125 (5×100/4).

So what numbers are the best ones? Lets assume several things.

1: Only touchdowns and field goals happen (ie, no safeties occur).

2: Touchdowns represent about 59% of all scores, field goals the rest.

3: Touchdowns are always worth 7 points (slightly unreasonable, touchdowns can be worth 6 or 8 points), and field goals are always worth 3 points (this one is true).

4: There are about 4 scores per team per game. (on average this is about right, but to truly do this you need not just the average information on the distribution of scores as well).

5: The score of one team has no effect on the score of the other team (again, this is false, a team trailing by 4 with 20 seconds to go, on a 4th and 20, is not going to kick a field goal, no matter what, a team trailing by 2 on a fourth down near the opponents end zone will absolutely kick a field goal).

6: The game can end in a tie (again, obviously false, and it should make all x-x numbers slightly less valuable)

The peculiar thing about football scores is that you can start a number line with 0, cycle through the numbers like this:

0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0

A field goal will move the active score one to the right, a touchdown one to the left. So a touchdown and a field goal will have no effect on the score. Then, all we need to do to get an estimate of the value of squares is as follows:

1: estimate the distribution of scores per quarter (I’ve set one up with a mode and average of 1 score per quarter, mostly normal distribution with a slight spike at 0 (to account for “negative” scores))

2: calculate the binomial distribution that within a given number of scores per quarter they will be touchdowns or field goals (so with 4 scores, 3 successes would be in excel =binomdist(4, 3, .59,false))

3: create a “tick” value for each scoring combination with field goals as +1 tick and touchdowns as -1 tick (so 4 scores, 1 touchdown would be a + 2 tick)

4: Multiply the probabilities in section 1 and section 2 together to get the probability of each scenario

5: Convert each “tick” value to its corresponding “point” value on the number line above, (so 0 becomes 0, 1 becomes 3, -2 becomes 4).

6: Total the probabilities associated with each point value

And you will get the probability of each end digit for the score for the end of each quarter. Multiplying them against each other gets the following grids:

End of First Quarter
0 7 3 4 6 1 9 8 5 2
0 21.05% 12.12% 8.37% 2.08% 0.99% 0.69% 0.23% 0.19% 0.09% 0.06%
7 12.12% 6.98% 4.82% 1.20% 0.57% 0.40% 0.14% 0.11% 0.05% 0.03%
3 8.37% 4.82% 3.33% 0.83% 0.39% 0.27% 0.09% 0.08% 0.04% 0.02%
4 2.08% 1.20% 0.83% 0.21% 0.10% 0.07% 0.02% 0.02% 0.01% 0.01%
6 0.99% 0.57% 0.39% 0.10% 0.05% 0.03% 0.01% 0.01% 0.00% 0.00%
1 0.69% 0.40% 0.27% 0.07% 0.03% 0.02% 0.01% 0.01% 0.00% 0.00%
9 0.23% 0.14% 0.09% 0.02% 0.01% 0.01% 0.00% 0.00% 0.00% 0.00%
8 0.19% 0.11% 0.08% 0.02% 0.01% 0.01% 0.00% 0.00% 0.00% 0.00%
5 0.09% 0.05% 0.04% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
2 0.06% 0.03% 0.02% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
End of Half
0 7 3 4 6 1 9 8 5 2
0 14.17% 8.09% 5.59% 4.12% 1.98% 1.96% 0.66% 0.64% 0.22% 0.21%
7 8.09% 4.62% 3.19% 2.36% 1.13% 1.12% 0.38% 0.36% 0.13% 0.12%
3 5.59% 3.19% 2.21% 1.63% 0.78% 0.77% 0.26% 0.25% 0.09% 0.08%
4 4.12% 2.36% 1.63% 1.20% 0.57% 0.57% 0.19% 0.19% 0.06% 0.06%
6 1.98% 1.13% 0.78% 0.57% 0.28% 0.27% 0.09% 0.09% 0.03% 0.03%
1 1.96% 1.12% 0.77% 0.57% 0.27% 0.27% 0.09% 0.09% 0.03% 0.03%
9 0.66% 0.38% 0.26% 0.19% 0.09% 0.09% 0.03% 0.03% 0.01% 0.01%
8 0.64% 0.36% 0.25% 0.19% 0.09% 0.09% 0.03% 0.03% 0.01% 0.01%
5 0.22% 0.13% 0.09% 0.06% 0.03% 0.03% 0.01% 0.01% 0.00% 0.00%
2 0.21% 0.12% 0.08% 0.06% 0.03% 0.03% 0.01% 0.01% 0.00% 0.00%
End of Third Quarter
0 7 4 3 1 6 8 9 5 2
0 9.92% 5.67% 4.62% 3.92% 2.36% 2.21% 1.16% 0.82% 0.43% 0.37%
7 5.67% 3.25% 2.64% 2.24% 1.35% 1.27% 0.66% 0.47% 0.25% 0.21%
4 4.62% 2.64% 2.15% 1.83% 1.10% 1.03% 0.54% 0.38% 0.20% 0.17%
3 3.92% 2.24% 1.83% 1.55% 0.94% 0.88% 0.46% 0.32% 0.17% 0.15%
1 2.36% 1.35% 1.10% 0.94% 0.56% 0.53% 0.28% 0.19% 0.10% 0.09%
6 2.21% 1.27% 1.03% 0.88% 0.53% 0.49% 0.26% 0.18% 0.10% 0.08%
8 1.16% 0.66% 0.54% 0.46% 0.28% 0.26% 0.14% 0.10% 0.05% 0.04%
9 0.82% 0.47% 0.38% 0.32% 0.19% 0.18% 0.10% 0.07% 0.04% 0.03%
5 0.43% 0.25% 0.20% 0.17% 0.10% 0.10% 0.05% 0.04% 0.02% 0.02%
2 0.37% 0.21% 0.17% 0.15% 0.09% 0.08% 0.04% 0.03% 0.02% 0.01%
End of Game
0 7 4 3 1 6 8 9 5 2
0 5.33% 4.31% 3.43% 2.98% 2.32% 1.65% 1.19% 0.82% 0.63% 0.43%
7 4.31% 3.48% 2.77% 2.41% 1.87% 1.33% 0.96% 0.66% 0.51% 0.35%
4 3.43% 2.77% 2.21% 1.92% 1.49% 1.06% 0.77% 0.53% 0.40% 0.28%
3 2.98% 2.41% 1.92% 1.67% 1.30% 0.92% 0.67% 0.46% 0.35% 0.24%
1 2.32% 1.87% 1.49% 1.30% 1.01% 0.72% 0.52% 0.36% 0.27% 0.19%
6 1.65% 1.33% 1.06% 0.92% 0.72% 0.51% 0.37% 0.25% 0.19% 0.13%
8 1.19% 0.96% 0.77% 0.67% 0.52% 0.37% 0.27% 0.18% 0.14% 0.10%
9 0.82% 0.66% 0.53% 0.46% 0.36% 0.25% 0.18% 0.13% 0.10% 0.07%
5 0.63% 0.51% 0.40% 0.35% 0.27% 0.19% 0.14% 0.10% 0.07% 0.05%
2 0.43% 0.35% 0.28% 0.24% 0.19% 0.13% 0.10% 0.07% 0.05% 0.03%

What can we learn?

The best numbers are 0, 7, 3 and 4, the worst are 5 and 2.

Numbers get more even the longer the game goes on, (this is born out by season ending statistics, which is essentially playing 64 quarters, as many teams have scores ending in 0 for the season (Giants with 380 and Arizona with 310) as have ending in 2, (Houston with 372 and Denver with 482)

Bad numbers are worse than good numbers are good. IE, 7-2 (one good number, one bad number), at the end of the game, is about as bad as 6-8 (two “ok” numbers). 0-2 (the best number and the worst number), is only marginally better than 1-9.

Even at the end of the game, only 33% of the numbers have positive expected payouts.

Pity to he who draws 2-2 (only .03% at the end of the game, which is possibly an overstatement, because the game cannot end in a tie).

Finally, I’d probably have done better to simply look at the scores at quarter end for each NFL game this year (or over multiple years) to get these probabilities, but I think my method has at least some value.