Astronomical BS

 A few years back, a friend and I got into an argument about astrology. She started telling me how much of a Gemini I am and how I very clearly match all the personality traits. As an experiment, I decided to screw with her and see how she reacted:  

Me: Actually, due to the precession of the equinoxes in the year I was born, I'm actually a Taurus. It's a very rare phenom. Most astrologists don't even know about it.  My birth year was the only time it's happened in the last 153 years.¹

Her: Oh my God!  That's so amazing!  I've always thought you acted like a Taurus.  Now I know why!

She then proceeded telling me how much of a Taurus I am and how I very clearly match all the personality traits.   I've repeated this experiment several times with different people, always with similar results.  No matter what sign I give, cognitive dissonance sets in, then there's rationalization, then I'm told how much of a [fill in the blank] I am and how I very clearly match all the personality traits.

Apparently, my little experiment was coincidentally prescient.  In light of the latest astrological buzz, it turns out I actually am a Taurus.²  Unbeknownst to people who follow astrology, the zodiac calendar has been wrong for years.  In addition to the various dates being off, there's also a 13th astrological sign that was left out.  How many astrology-believing Americans just had their signs switched?

Despite the absence of evidence or even a plausible mechanism³, many Americans still believe the stars control our personalities.  According to Michael Shermer's Why People Believe Weird Things, a 1990 Gallup poll showed that 52% of Americans believe in astrology.  There's very little overlap between the zodiac calendar and the actual positions of constellations in the sky. For this reason, about 90% of people's zodiac signs changed. This means that

(3.0×10⁸ Americans) · (0.52 believers per American) · (0.9 zodiac changes)
= 1.4×10⁸ zodiac sign changes.

That's 140 million people and a whole lot of cognitive dissonance.  Thank you, Laurie for suggesting this one.


[1] At this point, I was basically making things up and throwing in ad-libed technical jargon.  The actual "precession of equinoxes" has nothing to do with astrology.
[2] But, no, this does not justify my astrologist friend's assertion.  As mentioned earlier, it doesn't matter what sign I give, astrologers will always say my personality matches.
[3]  And, no, the fact that the moon controls the tides does not constitute a mechanism.  Tidal forces are due to gravity, and as I show in How Many Licks?, the tidal force on you at birth due to the obstetrician is about 40 times larger than the largest tidal force of any heavenly body.

Seth MacFarlane: Family Guy Creator or Math Genius?

On December 26, 2010, Family Guy creator and voice actor extraordinaire Seth MacFarlane tweeted, "If the T1000 had to kill everyone named Aiden Connor or Dylan Connor, he’d have tons of killing to do."  How accurate is his estimate?

According to Book Of Odds, each name has the following probabilities:

• The odds a male is named Dylan are 1 in 6,250 (US, 5/1990 - 9/1990). 
• The odds a male born in 2000 is named Aiden are 1 in 2,370 (US, 2000). 
• The odds a person's last name is Conner are 1 in 4,442 (US, 2000).

Assuming there are no weird correlations, this means the probability that a person is named "Dylan Conner" is 1 in 28 million and the probability that a person is named "Aiden Conner" is 1 in 11 million.  Just considering the U.S. population, there are 150 million males roughly 5 and 12 of which would be named "Dylan Conner" and "Aiden Conner", respectively.  If each Conner weighed 150 lbs, the total weight would be 1.3 tons.

Well done Mr. MacFarlane!  [In poorly attempted Stewie voice] Victory is yours!

Lessons from Stick Figures Presents, "A Brief Interlude on Entropy"

Plague Gauge

"This is what the LORD says: By this you will know that I am the LORD: With the staff that is in my hand I will strike the water of the Nile, and it will be changed into blood. The fish in the Nile will die, and the river will stink; the Egyptians will not be able to drink its water."

— Exodus 7:17–18

Eeeew!  How many people would it take to make a river of blood? 

According to the Wikipedia, the Nile discharges 2830 m³ of water each second.  You'd need an equivalent flow rate of blood to produce a Nile-sized river.  According to the American Red Cross, you can give 1 pint of blood once every 56 days. To produce a river worth of blood, you would need a total of

# of people = (flow rate required) / (flow rate per person)
= ( 2830 m³/s ) / ( 1 pint / 56 days )
= 2.9 ×10¹³ people.

That's 4000 times more people than there are in the world today.

2011

Unlike plants, cats, or small children, this numbers blog doesn't complain when I don't feed it for two months.  Still, I've been feeling bad about not posting, so my New Years resolution is to start writing regularly again.  More to come in a moment...

Death By Coconut

In “I Know What You Did Last Summer of the Shark”, then Daily Show correspondent Stephen Colbert states that falling coconuts kill 150 people each year. You might assume that “death by coconut” was a purely random occurrence, but that might just be what the coconuts want you to believe. What's are the chances that the coconuts are out to get you? 

According to Wikipedia, 54 million tonnes of coconuts were produced in 2009. From this, we know that if each coconut weighs 10 lbs, then roughly 1.0×10¹⁰ coconuts were produced. Now, we could quibble about the actual number. Some grow in the wild which might make the actual number larger. Some coconuts are picked instead of falling, so that might make the actual number smaller. You can argue either way, so let’s stick with this figure just to keep the problem relatively simple.

There are 6.7×10⁹ people in the world, each of which has about 1.5 ft² of area that the coconut could land on giving a total area of about 1.0×10¹⁰ ft².  According to Wikipedia, the total land area in the world is about 1.5×10⁸ km².  If people are randomly distributed across the land area of Earth, then the probability of being hit by a coconut is equal to the fraction of land area that people take up at any given time.  Using Google’s calculator, we get

From this, we suspect that each year roughly 6 out of every one million people get bonked by a coconut.
It’s difficult to say how many people actually get killed by coconuts. From our calculation above and the world population, we can estimate the number of people that get hit by coconuts each year, but not everyone who gets hit from a falling coconut will be killed by it. Assuming all hits were fatal, we could calculate the total number of deaths by multiplying “hits per person” times the “total number of people”, to get

(6×10^-6 hits per person) × (6.7×10⁹ people) = 40,000 fatal hits.

If hits are only fatal 1% of the time, then 400 people will die from coconuts falling. However, this 1% statistic may be off substantially—possibly much more than an order of magnitude—so our estimate is very rough.

It’s difficult to tell how many coconut-induced fatalities will occur, but our estimate of “40,000 coconut hits” suggests that the number of deaths could be substantially higher than 150. If this were the case, then coconuts are certainly not out to get you since they kill fewer people than they would by chance.

We’ve assumed the following:

· Coconuts are 10 lbs on average.
· The mass of all the falling coconuts in the world each year is equivalent to the mass of coconuts produced each year.     
· The probability of a falling coconut hitting a person is equal to the fraction of land area taken up by people.
· Only 1% of coconut hits are fatal.

These assumptions seem reasonable, but that does not mean they are necessarily correct. Coconuts are certainly between 1 and 100 lbs, so the first assumption seems decent. It’s possible the number of falling coconuts each year is off for the reasons stated above. Likewise, the percentage of fatalities could be off by several orders of magnitude. Equating the probability of a falling coconut hitting a person to the fraction of land area taken up by people is a reasonable first guess, but there are factors that might throw this assumption off. For example, perhaps more people live near coconut trees because people like to live in tropical climates. 
 
While our estimate doesn’t have enough precision to answer this question conclusively, this example does illustrate an important point. As Weinstein and Adams describe in their book Guesstimation, estimates generally break up into three “Goldilocks” catagories: too big, too small, and just right. In this case, being “just right” means your estimated result is too close to call. When this happens, you need to put more effort into refining your estimate if you want to draw any conclusions. Refining the coconuts estimate to high precision is beyond the scope of what I can do in a silly blog post, but there is still a valuable lesson to be learned: In estimation as in life, there are times when even the best answer leaves a wide degree of uncertainty and it’s important to acknowledge when we don’t have enough information to draw a conclusion. That said, there are many examples where a test produces results that are so unlikely we can conclude they are not due to random chance.

Lucky Numbers

Anna and I went out for Chinese food in Philadelphia today. As I looked at the lucky numbers in my fortune cookie, I couldn't help but wonder, "If everyone who ate Chinese food today played their lucky numbers in the lottery, what are the chances at least one of them would win?"

Both fortune cookies and lottery numbers usually show about 5 numbers that can range from roughly 1 to 50.  The probability of picking the first number correctly is 5 out of 50.  The probability of picking the second number correctly is 4 out of 49.  The probability of picking the third number correctly is ...  Multiplying these probabilities together, we can find the total probability of finding the right sequence of numbers¹,

P = [(5)! · (50-5)!] / 50! = 4.7×10^-7.

That's about one in two million. I generally go out for Chinese food about once per month, which seems like a reasonable amount for most people.  Taking that as the average and using the fact that there are 3.0×10⁸ Americans, we can estimate the number of people that went out for Chinese today,

# of people going for Chinese = (prob. of going out for Chinese) · (total # of people)

= (1 day / 30 days) · (3.0×10⁸ people)

= 1.0×10⁷ people.

The probability that everyone will will pick the right numbers is P^10,000,000.  Likewise, the probability of everyone picking the wrong number is (1-P)^10,000,000. The probability that at least one person will win is then just

1 - (1-P)^10,000,000

= 0.009.

There's about a 1% chance that if everyone played their lucky fortune cookie numbers at leat one would win.

[1] This is the well known binomial distribution.
[2] I'm assuming the fortune cookie's "lucky numbers" are random and uniformly distributed.