Archive for December, 2008

Carnival of Mathematics #45: The Procrastination Edition

December 5, 2008

clown at the carnivalIt’s time for the 45th Carnival of Mathematics! It’s being hosted by The Teaching College Mathematics Technology Blog, and Maria there has thoughtfully grouped the posts into categories of what they will help you procrastinate.  I was a little embarrassed at just how many of my current procrastinations she pinpointed, but I’ll forgive her because this Carnival has so many good posts!

The Final Stretch

December 4, 2008

Classes are finishing up, projects are graded, and finals start next week.  To celebrate the end of the semester, one of our majors brought in a mix of math songs.  Only most of them weren’t initially about math, they just sounded that way after her editing.

In honor of this week’s busyness and for math songs everywhere, here’s a video of one of the songs that Jill used.  It’s “U + Me = Us (Calculus)” from 2ge+her:

[And if you’re just feeling depressed about the mathematical subject, you can find the parody lyrics “Don’t Know My (Calculus)” here.  Although a parody of a satire is kind of a funny phenomenon.]

We’ve Been Nominated!

December 3, 2008

The polls are open for the 2008 EduBlog Awards, and 360 has been nominated in the “Best Group Blog” category!  Thank you to Maria and Mike and anyone else who nominated us!  Head over and vote in up to 16 categories.

Proceed with Caution

December 2, 2008

Deck of cards

I ran across an interesting math mistake in a post by Bob Murphy on the blog Crash Landing. He quotes from Peter L. Bernstein’s 1993 book Capital Ideas (which is apparently “A savvy appreciation of how a small band of disinterested academics has revolutionized the way Wall Street and its offshore counterparts manage the world’s investment wealth,” according to Kirkus Review).

Anyway, here’s the quote:

[Cowles] must have been a fiendish bridge player. Here is one passage from his notes on the game:

If each of 50 million bridge players in the US plays 200 sessions of 40 deals each, this adds up to 50 million*200*40 = 400 billion hands dealt each in US (sic). The probabilities on any given hand being dealt with 13 cards of one suit are .00000000000156. The chances of a hand with 13 cards of one suit being dealt in the US in any given year, therefore are 400 billion times .00000000000156=.624.

The challenge posed was to find the math mistake. I’m pretty sure I know which one he was referring to, but I’ll share my train of thought anyway.

My first guess was that it has to do with the 400 billion deals going on in the US each year. But in retrospect, I think this number is correct given the assumptions [and assuming that it’s only the dealer’s hand that’s being looked at; there are four people playing with each deal]. What I disagree with is that those assumptions are reasonable. Are there really 50 million bridge players in the US? Maybe. But the US Census estimates that there are only 301,621,157 people in the US as of July 1, 2007; that means that 1 in 6 people is a bridge player. It might be true that 1/6 of the folk in the US know how to play bridge (maybe), but I’m pretty sure that it is not the case that 1/6 of the entire US population deals 8000 hands per year. That’s almost 22 hands per day (and that only counts when that person is the dealer!), each and every day, for an entire year. If that were the case, who would have time for blogging?

So that 400 billion is way off, but because of unreasonable assumptions rather than an actual mistake (and not having read the actual book, it’s possible he knew that he was wildly overestimating the number of hands dealt). Then I turned my attention to the line “The probabilities on any given hand being dealt with 13 cards of one suit are .00000000000156.” This number comes from the fact that the number of ways to choose 13 cards from a deck of 52 is (52 choose 13), or approximately 6.35×1011. There’s only one way to get all 13 cards in a particular suit (say, spades) so the chances of getting that or any particular configuration of cards is 1/(52 choose 13) or approximately 1.57×10-12. So that number fits the paragraph above, if you interpret “a suit” to mean a particular suit. If you just want all the cards to be of the same suit, then you’d have to multiply that probability by 4.

Which means that probability is probably wrong too, but it does depend on interpretation. Finally I turned my attention to last line, about multiplying the probability by 400 billion to find the likelihood of getting all cards in the same suit because there are 400 billion hands dealt. And this, I am certain, is the error that Bob Murphy was referring to. As Bob’s brother pointed out, if they had used a population that was large enough (or indeed, simply multiplied the tiny probability above by 4 to take into account that there are 4 different suits in the deck), the probability of getting all 13 cards in the same suit would appear to be more than one.

The correct way to solve this would be to find the likelihood that a particular hand dealt was not all of a given suit (1-1.57×10-12), raise that to the 400 billion power to find the probability that none of those 400 billion hands dealt were all of a suit, and then subtracting that from 1. Using Excel, it appears that there’s a 99.996% change that someone, somewhere got that particular configuration of cards.

What a rich little paragraph! It looks like Bernstein has written a sequel to the book in 2005; let’s hope that in this version, the math was checked a bit more carefully.

Math in Bones

December 1, 2008

dvdSo I’ve been watching DVDs of Bones, as you might have gathered from an earlier post. And one of the episodes from Season 2, “Spaceman in the Crater”, has three separate spots with some math! Not a lot of math, but it doesn’t take much to make me happy.

The first bit comes early on, when Special Agent Booth and Forensic Anthropologist Dr. “Bones” Brennan examine a body that’s fallen from the sky and formed a crater.

Brennan (immediately after commenting on the man’s loafers): He hit the ground at approximately 200 kilometers per hour.
Booth: How can you tell that by his shoes?
Brennan: 124 miles per hour is terminal velocity for a falling human.

See that fancy unit conversion? It’s right, too. (OK, not hard math, but for all we know Brennan used Fibonacci numbers to figure it out, since 200=144+55+1 [i.e. F12 + F10 + F2] so the miles would be approximately 89+34+1=124 [i.e. F11 + F9 + F1].

Then a little later, Dr. Zack Addy took it a bit further:

A human being reaches terminal velocity after falling 200 to 220 meters, depending upon air resistance. Velocity would be achieved between 5 and 8 seconds, depending upon atmospheric conditions, body position, and clothing. He fell from a minimum of 1200 feet. I can run through the math if you like.

Sadly, no one wanted Addy to run through the math so I can’t see what played into this. Presumably if air resistance were negligible (which it isn’t) and if the body was dropped rather than pushed, then it would fall ½(9.8)t2 meters after t seconds; that’s 122.5 meters per second after 5 seconds and 313.5 meters per second after 8 seconds. The upper bound of 8 seconds makes sense to me, since air resistance would slow the speed down, but I’m a little surprised by the lower bound of 5.  Shouldn’t it be at least 6 seconds, and probably closer to 7?  Maybe Addy is thinking the body might have been pushed.

In the final math segment there’s a bit of exponential decay. Addy shares the following observation a bit later, after they’ve determined that the victim was an astronaut:

Astronauts lose 2% of their bone mass for each month spent in space. Our victim’s legs, hips, and lower vertebrae have demineralized over 20%, indicating 10 months in space.

Using that 2% per year, the amount of bone mass left should be (0.98)n after n months. Solving (0.98)n=0.8, in order to find out when the boned demineralized 20%, leads to n=log(.8)/log(.98) [where the log is to your favorite base], or n≈11.04 months. That doesn’t fit with the 10 months mentioned above. In this case, though, rounding could be the culprit: if it was really 2.4% of bone decay per month and that number was rounded down to 2% for convenience, the formula becomes (0.976)n=0.8, giving n=log(.8)/log(.976)≈9.2, which fits with the data. I’ll grant Addy this one.

(And yes, the rest of the episode was good too!)