The joke ran at the end of a muppet-themed segment of the show.  In an homage to Sesame Street, after the segment finished (but before the somewhat racy joke involving a very physical muppet Moe) an announcer stopped to give thanks to the sponsors of the show.  Unlike Sesame Street, however, which is sponsored every day by two letters and a number, this episode of The Simpsons was sponsored by one symbol and one number that looks like a letter:

In case you’re late to the party (since I don’t think that clip will be online forever), let me quote: “Tonight’s Simpsons episode was brought to you by the symbol umlaut, and the number e. Not the letter e, but the number, whose exponential function is the derivative of itself.”

Kudos to the writers for incorporating some choice math humor into the tail end of this episode (I’m willing to overlook some qualms with their wording).  Perhaps Simpsons aficianados would can begin preparations for next year’s e day.

As a concrete example I considered the case N = 6, M = 5, K = 3 – in this case, the minimum number of questions you should prepare to guarantee that you can answer 3 of 5 problems on the exam is 4, and we saw that if you only prepare 3 questions, you have a 50% chance of those 3 questions appearing on the list of 5.

Late last week, however, I was made aware of another example, one for which the probabilities might prove more interesting (since there are more cases to consider).  Specifically, let us consider the case of a person studying to become a U.S. citizen.  As part of this process, one must submit to an interview in which one is asked 10 questions, and must answer 6 of those 10 questions correctly.  However, the potential list of questions is made available to people beforehand; there are 100 questions from which the 10 questions can be drawn.  In other words, we have N = 100, M = 10, and K = 6.

In this case, to guarantee that you will be able to answer 6 of the 10 questions presented, our analysis from last time tells you that you should prepare 100 + 6 – 10 = 96 of the questions.  Indeed, this makes sense, since the worst that can happen is that the 4 questions you don’t prepare happen to be precisely 4 of the 10 questions you are asked in the interview.  This also reflects the fact that the closer M is to K, the more questions the test taker will have to prepare (note that if M were closer to N, say M = 90, the test taker would only have to prepare 16 questions).

Still, preparing 96 of the questions may seem like a little much, especially since only 10 questions will come up in the interview.  So, let’s see what happens if someone prepares for fewer than 96 questions.  Obviously one should know how to answer at least 6 of the questions, but what about values between 6 and 96?

Here is a graph showing the probability that one will pass the interview given that one has learned the answer to n questions, for some n between 6 and 96.This graph tells you that, for example, even if one only had time to learn the answers to 73 out of the 100 questions, one’s chances of passing the exam would still be over 90%.  Those are pretty good odds, for only learning the answers to roughly three quarters of the questions.  On the other hand, one needs to learn the answers to 37 questions before one’s odds of passing rise above 10%, so it’s certainly not likely that someone will pass by learning the answers to only a handful of questions (which is probably what the government intends).

If only Apu had known of these findings, perhaps he could have saved himself some trouble.

Anyway, I just wanted to highlight another example where these ideas apply.  If you can think of any others, let me know!  Also, if you are interested in the content of the 100 questions that can be asked of our future citizens, you can find the full list (along with acceptable answers) here.

2 years seems like enough time for such an unequivocal conclusion to have begun seeping into our collective consciousness, but apparently not. I welcome the day when I am no longer asked to form separate lines while buying my lunch – it’s like asking me to give you even more of my (quite valuable) time. It’s not that we can’t hear you, cashier, it’s that we know what’s in our own best interest. And frankly, so should you.

Is there any advantage to using multiple lines? There may be some psychological benefit to having many short lines rather than one longer line, especially for people who, for example, may go to the grocery store only to pick up one or two items. For them, the sight of a single line may be overwhelming, even if that line does move much faster than separate shorter lines would.

Also, there is perhaps something to be said for the use of express lines, which cater to those people who would be most turned off by a long line. However, with the single line system, that one line is already express! Neither of these points seem to matter much in the face of the data, which strongly points to having your customers stay put in a single line.

So, the next time you’re waiting in line, and the cashier asks you to split into smaller lines, feel free to hold your head up high and ask what’s in it for you. You certainly aren’t doing yourself any favors by complying.