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.

Special Guest Natalie Angier

Today we're pleased to have a question from special guest Natalie Angier.  Ms. Angier is a Pulitzer-prize winning science journalist for the New York Times.  She has authored several books, most recently, The Canon: A Whirligig Tour of the Beautiful Basics of Science.  She writes, "How many leaves are raked up nationwide on an average weekend afternoon in October?"


Before I begin, I have leaf-raking riddle for you.  Without adding or rearranging the words, add punctuation to the following sentence to make it grammatically correct: "A boy raking leaves." The answer is below.

If you're going to rake leaves, you need leaves to rake.  Some states like Arizona are desert-y and won't have many leaves to rake, but most places in the U.S. will have trees that shed.  Even if you live in the right climate, you still need a yard with at least one tree in it.  I'll assume that 1 out of 10 people owns a yard with a tree in it, since it's very likely that the actual number is greater than 1 out of 100 and less than 1 out of 1.  Of these people, some will rake, but many will use a leaf blower or just let the leaves lie.  Using a similar order-of-magnitude argument to the previous one, I'll assume 1 out of 10 people who have leaves rake them.  The average leaf raker might rake his/her leaves once in October, and there are about 8 good leaf-raking October weekend afternoons each year.  There are 3.1×10⁸ people in the United States.   Combining these assumptions, we can estimate that,

# of people raking = (3.1×10⁸ people) · (0.1 tree owners per person)
· (0.1 raker per tree owner)· (0.125 chance of raking now)

= 390,000 people raking leaves each weekend afternoon in October.

But the question specifically asked for the number of leaves raked.  This will clearly depend on the number of leaves a person has in his/her backyard.  According to at least one source, a mature tree can have up to 200,000 leaves.  To confirm this, I looked at leaves strewn across Tappan Square in Oberlin.  The mean separation was about 6 inches between leaves.  If you spread them out over a reasonably sized lawn (about 1/5 of an acre), you get about 200,000 leaves.  If each raker rakes this many leaves one a weekend, there will be,

# of leaves = (200,000 leaves per raker) · (390,000 raker)

= 7.8×10¹⁰ leaves raked.

That's 78 billion leaves raked nationwide each afternoon in October.  Thanks for the great question, Ms. Angier! 

For those wondering about my earlier grammatical riddle, the correct answer is "A boy, raking, leaves."

A Relatively Good Calculation

In 1905, Einstein published his special theory of relativity.  The most well-known part of this theory is almost certainly the famous E=mc² equation that predicted a future with nuclear bombs and atomic energy, but this is not the only surprising prediction.  The theory also predicted that objects shrink when they move really fast.¹  After hearing a professor describe this strange and fascinating phenomenon, I wondered two things.   First, what was Einstein smoking?  Second, if I ran really fast, would I be able to see atoms?  How fast does a person have to run to be atom sized?

According to special relativity, the length of a moving object is equal to its original length times an extra factor

L' = L [1 – (v/c)²]^0.5.

Here, c is the speed of light, L' is the length of the object when it's moving, L is the length of the object when it's not moving, and v is the velocity at which it's moving.  Our new length L' will be about 10^-10 m or roughly the size of an atom.  Our original length will be about 1.5 m.  We can solve for v

v = c [1 – (L'/L)² ]^0.5

 = (3.0×10⁸ m/s)[1 – (10^-10 m / 1.5 m)² ]^0.5

= 0.9999999999999999999977778 c.

You would need to move very close to the speed of light to be atom sized.

[1] This phenomenon is called "length contraction".

Lieutenant Commander Banana Clip

I saw this on Totally Looks Like the other day, and it certainly brought back a few repressed memories.  When I was a kid, I definitely wore a banana clip pretending to be Geordi LaForge, and my family wasn't even big Star Trek fans.  How many people in the U.S. wore banana clips over their eyes pretending to be Lieutenant Commander Geordi LaForge?

Apparently, I was not alone.  It strikes me that role playing as Geordi takes three things: a certain level of maturity, Star Trek knowledge, and banana clips.  I chose the phrase "a certain level of maturity" carefully.  I'm pretty sure most 2-year olds and 90-year olds didn't do this.  You need just the right amount of maturity.  With too much, you just think it's silly.  With too little, you can't see how genius the banana clip Geordi visor really is.  I suspect about 1/3 of the population has the appropriate maturity level. 

After maturity, you need Star Trek knowledge.  As I said earlier, I wasn't even that big of a Star Trek fan, but I still rocked the Geordi visor.  That said, I suspect at least 1 out of 10 people had my level of Star Trek knowledge. 

The final ingredient is, of course, banana clips.  These were fairly ubiquitous in the late 80s and early 90s when Star Trek the Next Generation was popular.  As a reasonable guess, I would say at least at least 1 out of 10 people had access to banana clips at this time.

Using the assumptions above, we can estimate that about 1 out of every 300 Americans, or about one-million people pulled off the Geordi look.  LeVar Burton, you deserve a commission from the banana clip companies.

[1] Doh!  "Jordi LaForge" should be spelled "Geordi La Forge."  Thank you, to my student Kara Kundert for the correction!