Thursday, September 30, 2010

Safety in Numbers

It’s commonly said that airplanes are the safest mode of transportation. It’s true that more people die in car crashes than plane crashes each year, but most people also drive more often than fly.  On a "per trip" basis, which mode of transportation is safer?

The question did not specify whether we were considering just crashes in the U.S. or in the entire world.  Assuming crashes are equally likely in all parts of the world, the “fatalities to trips” ratio should be about the same in both cases.  Just make sure you are consistent (i.e. don’t divide the number of U.S. plane crash victims by the total number of flights in the world.)

According to “Ask a Scientist” 1, in the U.S. there are

“…about 40,000 deaths per year in automobile accidents vs. about 200 in air transport.”

You can check these numbers against other references to make sure they are accurate.  To answer the question, we need to estimate how often a person flies and how often a person drives.  On average, we might guess that an American flies about once per year.  This is reasonable since you’d certainly expect people to fly more than once every 10 years and less than once per month.  In contrast, most of us drive (or ride in) a car about twice per day, even if it’s just to get to and from work or school.  Since there are 3.1×108 Americans, this means there are

# flights per year = (1 flight per American per year) × (3.1×108 Americans)
= 3.1×108 flights per year,


# car trips per year= (2×365 car trips per Amer. per year) × (3.1×108 Americans)
= 2.2×1011 car trips per year.

The fraction of deaths is then just,

Fraction = (# deaths per year) / (# trips per year)
= (200 deaths per year) / (3.1×108 plane trips per year)
6.5×10-7 deaths per trip


= (40,000 deaths per year) / (2.2×1011 car trips per year)
1.8×10-6 deaths per trip.

You are 2-3 times more likely to die in a car trip than a plane trip, so according to our numbers plane travel is still safer.  Given the precision of the estimation, it’s possible that other reasonable assumptions would come up with the opposite result, since both figures are within an order of magnitude of each other.

[1] You can find this stat and a nice discussion of the topic at here.  Their numbers are different partly because they’re talking about fatalities per mile not fatalities per trip.

Sunday, September 26, 2010

I Can Haz Estimation?

If you’re like me, there’s nothing you like better after a long work day than kicking back and looking at some grammatically incorrect cats.  And who better to provide all your LOLcat needs than The Cheezburger Network?  In honor of my favorite intertubes diversion, how many pictures of cats are on the entire Internet?

This is a difficult problem.  That being the case, we expect our estimate will likely be very different from the actual number.  In cases like these, it is helpful to determine what the wrong answers are.  To do this, you need to calculate upper and lower bounds.  To start, let’s estimate how many people are cat owners.  You might guess that 10% of people own cats.  Is this reasonable?  Well the actual number is certainly less than 100% of people, and very likely greater than 1% of people.  If you asked a 100 of your friends and family, at least one of them probably has a cat.  The next question you could reasonably ask is how many cats do cat owners typically have.  Some people have 10 cats, while others only have one.  A reasonable guess is 2 cats per owner. 

Now comes the tricky part.  What fraction of people put pictures of their cat on the web?  Many of you are tech savvy and have accounts on Facebook, Flickr, etc.  If you have a cat, you very likely have at least one picture of it on the web somewhere.  But what about your parents, grandparents, and friends that may not be as tech savvy as you?  What about people in developing nations that may not have Internet access?  If you average over everyone including people in other countries, what percentage of cat owners will post pictures on the web?  To be safe, lets say 10% again.  Is this reasonable?  As before, the actual number will certainly be less than 100%.  Will it be great than 1%?  Possibly, but there are a lot of people that don’t like putting their information up on the Internet and the ones that do might not put their cat on the Internet.  To be safe, let’s put the lower bound at 0.1%.  We can summarize what we know in a chart like the one below:

% of people that own cats
Cats per owner
% of owners that put cat on web
Pictures per cat
World population
# of cat pictures

Our upper and lower bounds are very far apart (i.e. 9 orders of magnitude).  The answer certainly lies between them, but is there any way we can tighten these bounds?

Let’s try calculating the answer in a different way.  According to Netcraft, there are 2.5x1010 web pages as of a few years ago. As an upper bound, we might say that at most, every page has 10 cat pictures on it.  Even in this extreme case, there would only be 2.5x1011 cat pictures, so we’ve effectively reduced our upper bound by a factor of 100. 

Now lets see if we can adjust our lower bound.  A quick Google image search for the word “cat” pulls up about 1.5x108 results.  Looking at the pictures on the first page, we can see that the vast majority of the images are of cats, but occasionally you get one that’s not a cat.  This is very rare.  In fact, it seems much more likely that Google is missing cat pictures since it doesn’t have access to private photos on sites like Facebook and Flickr.    For this reason, we could probably take the Google result as a lower bound.  Just to be extra careful, let’s say the lower bound is 1/10th the number of “cat” images found by Google.

Our final results:

Upper Bound -- 2.5x1011 cat pictures
Actual Estimate – 1.3x109
Lower Bound -- 1.5x107

Tuesday, September 21, 2010

My Eyes Are Buggin’

I had an interesting conversation with one of my physics students recently.  She had a dance class right before she came to lab.

Her: I hate dancing with glasses.

Me: Because they fall off when you’re doing the turn-y things?

Her: Yes, it’s so much easier with contacts.

Me: You know, you can calculate how fast you’d have to turn to get the contacts to pop out.

How fast would a dancer have to spin to get her contacts to pop out?

Contact lenses weigh about 0.05 g and are about 1.0 cm2 in area.  They are held on mostly by suction.  Suction occurs because there is a partial vacuum created under the contact as it's pulled away from the eye.  To pop out, a contact needs to overcome air pressure, which for our atmosphere is about 100 kPa.  As the dancer spins, her contacts move in a circle with roughly a 10 cm radius.  The force to keep an object in uniform circular motion is given by

force = (mass) · (velocity)2/ (radius).

The maximum this force can be before the contact pops out is determine by atmospheric pressure

force = (pressure) · (area).

Solving for the velocity, we get

velocity = [(pressure) · (area) · (radius) / (mass)]1/2
= [(100 kPa) · (1 cm2) · (10 cm) / (0.05 g)]1/2
= 140 m/s.

A dancer would have to rotate about 1400 per second to get her contacts to pop out.

Smile! You’re the Monocle Smile Contest Winner!

Hello, Ladies.  How are you?  Fantastic.  We have a winner!

The Question: If he worked nonstop for the rest of his life, how many personalized Youtube videos could Mr. Mustafa make?

Born February 11, 1974, Isaiah Mustafa is currently 36 years sexy.  He’s young and he’s in shape, the latter of which makes longevity more likely.  As best I can tell, he only has one major thing going against him: he played NFL football.  However, his career was fairly short and he was a wide receiver, so he’s not as likely to have the same health issues as a 12-year 350 lb lineman.  One simple way to solve this is to subtract his current age from his life expectancy.  According to Wikipedia, Americans are 38th in the world for life expectancy with and average lifespan of about 78.2 years.  Using Wolfram Alpha, we know that Mr. Mustafa was 36.6 years old on the contest deadline.  From this we can compute how many years he has left,

78.2 years – 36.6 years = 41.7 years.

Each personalized video appears to be about 30 seconds in length.  If he worked nonstop1 making 30-second videos for 41.7 years, he would have a total of about 44 million videos.

Congratulations to our winner, Joey Reichert!

[1] Some might argue that we haven’t taken into account sleeping.  True.  If he did sleep, our final number would only be one third as big.  But the original question did say “nonstop.”  “But wouldn’t he die younger if he never slept?” you ask, suspiciously.  Also true.  But we could always film him while he slept, and which of you ladies out there wouldn’t appreciate that?

Tuesday, September 14, 2010

A Cure for Global Warming?

I came across the above photo on Reddit the other day (original link here).  As one Redditor so aptly put it, “The stupid, it burns!!!”  Interestingly, there are multiple layers of stupid here.  Like a giant stupid onion.  Leaving aside the fact that water dumped into a sink would eventually wash back into the ocean, how many buckets of water would each Earthling need to dump to cancel out the rising sea levels?

According to the Wikipedia entry for "Current sea level rise", predicted values for the ocean rise due to global warming range anywhere from 90 to 880 mm.  I'll take 90 mm and just calculate the lower bound.  The total area of the oceans is roughly 3.4 ×1014 m2, meaning that the total volume is about 3.0×1013 m3.   One bucket might hold 50 L and there are 6.7 ×109 people in the world.  Form this, we can compute the total number of buckets needed,
# of buckets = (total volume) / [ (volume per bucket per person) × (number of people)]
  = (3.0×1013 m3) / [ (50 L per bucket per person) × (6.7×109 people)]
= 90,000 buckets.

Even neglecting the fact that the water will just wash back into the ocean, this method would still require every person on the planet to dump 90,000 buckets of sea water down their sink.  As Shakespeare might once have said, "The stupid, it doth burn."   

Friday, September 10, 2010

Bounding Beck

Sarah Palin said there were “hundreds and hundreds of thousands of people.”  Glenn Beck put the estimate at 500,000 people.  Congresswoman Michelle Bachmann stated that she wouldn’t let anyone tell her there were less than 1 million people. estimated only 87,000 people based on aerial pictures it took.  How many people were at the Glenn Beck rally?

As you can see from the photos on, most of the people assembled along the Lincoln Memorial Reflecting Pool.  According to Wikipedia, the pool is approximately 618 m long by 51 m wide.  Judging from the photos, attendees seem to cover an area between 3 and 4 reflecting pools wide, or roughly between 150 m and 200 m.  Multiplying by the reflecting pool length, this means the total area taken up by the attendees is between 92,700 m2 and 125,000 m2.  To be sure we’re not forgetting anyone, we’ll count the attendees that spilled over near the Washington monument and those that got a front row seat.   This looks like no more than 20% of the total number of people, so we’ll say they cover a total area of at least 90,000 mand at most 150,000 m2

Looking at the aerial photo, there’s a whole lot of green grass you can see between people.  This makes sense since most people don’t like to be squished and you need some extra room so that people can move around.  Each person needs at least 1.0 m2 of space to feel comfortable.  Judging from the sparse parts of the crowd, there’s appears to be no more than 10 m2 of space between people on average.  The average density certainly lies between these two extremes.  From these densities, we can use the equation

number of people = (density of people) × (area)

to compute an upper bound,

number of people = (1.0 person per m2) × (150,000 m2)
= 150,000 people

and lower bound,

number of people = (0.1 people per m2) × (90,000 m2)
9,000 people.

As you can see, the “87,000 people” estimate from AirPhotosLive lies right in the middle of the two extremes.  It is certainly a reasonable estimate, and it brings up an important point.  One way to test the accuracy of your estimate is using upper and lower bounds.  By using the largest and smallest realistic numbers in your estimation, you can put a cap on what the actual number must be.  We know there’s certainly more than our lower bound of 9,000 people and certainly less that our upper bound of 150,000 people. 

By using upper and lower bounds, we also see that our politicians and pundits are either lying or possess a certain degree of innumeracy. Of these two options, I sincerely hope it’s lying.  I’ve grown accustomed to politicians bending the truth, but I’m absolutely terrified by the thought that people who have gained this much influence can’t do middle school math. This does beg one to ask, “Why even bother lying about numbers like this?”  For one, false claims like this are easy to debunk even with mediocre math skills.  In addition, 87,000 people at a rally is nothing to sneeze at, so why exaggerate? 

Finally, I should note that I had some hesitation when writing this post.  This is not intended to be a political blog, and I’m not here to spout my personal beliefs.  The fact is that no matter what side of the political spectrum they’re on, rally organizers tend to exaggerate numbers to make it seem like they’re more popular than they really are, and it’s our duty as rationally thinking citizens to call them on it.