A Jumbo Margin of Error

I was out in the neighborhood shopping for seafood and stopped by the local fishmonger. I inspected the jumbo shrimp, which were going for $17.99 a pound. I decided I’d take a pound, and placed my order.

The fishmonger threw a handful of shrimp on the scale. The pile weighed in 0.85 pounds, so he added a few more. The additional shrimp took the total over a pound, so he took one off. Slowly, this series of shrimp was converging to the appropriate limit.

Still a little over, the fishmonger exchanged two large shrimp with two smaller ones. This brought the total weight down to 0.95 pounds.

I turned my attention to the rest of my grocery list. It wasn’t until after I had cashed out and left the store that it hit me: The fishmonger had charged me $17.99 for the shrimp! That is, he charged me for a full pound, even though I only received 0.95 pounds.

Now, a 5% margin of error might not seem too bad, but because of the high cost of shrimp, that 5% error amounts to 90 cents! With all the attention paid to that weighing, I feel like he could have been a bit more accurate.  I would also hypothesize that the vast majority of weighing inaccuracies are of the “under” variety.

My inattentiveness here cost me a dollar, but at least I walked away with something to think about. And while the fishmonger may have won a dollar, he lost all my future business.

Un-Random Shufflers

This is a great story about how statisticians at Stanford audited a new automatic shuffling machine and determined that the cards weren’t distributed randomly enough.

https://www.newscientist.com/blogs/onepercent/2011/07/shuffling.html

If a deck of cards is dealt one at a time, a knowledgeable observer, in theory, should be able to predict the next card dealt around 4.5 times per 52-card deck.  For example, by remembering which cards have been dealt, the observer will definitely know the final card, as it’s the only one that hasn’t been dealt.  Similarly, the observer will have a 1 in 2 chance of guessing the second-to-last card, and so on.  Calculations involving probability and expected value will give you the theoretical result.

For this particular shuffler, however, the statisticians from Stanford determined that an observer should be able to predict the next card 9.5 times per 52-card deck!  The shuffling machine manufacturer that hired them must have been pretty upset to hear this, but redesigning the machine is probably not as costly as selling casinos hundreds of predictable shufflers and then dealing with the consequences.

It should come as no surprise that Persi Diaconis is the lead author on the paper.  Diaconis is a living legend in the world of mathematics, having left home at an early age to become a sleight-of-hand artist, then returning to earn a PhD from Harvard in mathematical probability.  One of Diaconis first major results was proving that seven shuffles are necessary to “randomize” a standard 52-card deck.

The full paper from Stanford can be found here:

http://statistics.stanford.edu/~ckirby/techreports/GEN/2011/2011-08.pdf

www.MrHonner.com

Math and Art: Math and Computer Animation

This is a clear, concise, and fascinating overview of how some very advanced mathematical ideas are making their way into 3-D animation.

http://www1.ams.org/samplings/feature-column/fcarc-harmonic

Here’s the basic setup.   In order to efficiently model a character, you approximate it with a frame that is built around a few important points.  To move the character, you focus on moving just those points that define the frame.  Thus, moving the character from point A to point B boils down to understanding where those handful of crucial points go.

The tricky part is figuring out a way to smoothly bring all those in-between points along for the ride, and that’s where the math comes in.  The secret is to think of those in-between points as averages of the points that define the frame.  The article explains how barycentric coordinates, harmonic functions. and a surprising amount of calculus are being used to pull off this movie magic!

Yet Another Car Rental Scam

As a car-less person in a car-driven world, I occasionally need to rent a car to move sofas, bring home large plants, or just get away from all my car-less neighbors for the weekend.  A long history of distrust of car-related business people (sales, repair) makes me approach my car-rental interactions with cynicism.  Here’s yet another reason that cynicism is justified.

Car rental agencies offer a Pre-Pay for Gas option when you pick up your vehicle.  “Pay now, and you won’t have to worry about filling the tank when you return the car.  It’s less hassle for you! ” says the salesperson.  “You even get a discounted rate on gas!”  It’s not emphasized initially, but you won’t be charged what it costs to fill the tank when you return the car; you will be charged for a full tank of gas, regardless of how much gas is in the tank.

This is actually a good deal for you if you can arrange your travel in such a way as to roll into the lot on an empty tank of gas.  But if you pre-pay and then return the car with half a tank of fuel?  You just gave the rental agency 8 gallons X $4 per gallon = $32 worth of gas.  That’s a nice little bonus for them!

And for the record, the sort of meticulous, obsessive planning required to return a rental car on empty tends to detract from the fun of your road-trip.  Not for me, necessarily, but potentially for your passengers.

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