Derangements

A couple of weeks ago I was at the Millbrae BART station and a young man asked me if I knew how to get to Slim’s in San Francisco. I didn’t, but I had my iPhone on me so I used Google Maps and showed him how to get there. After we got on the train we chatted some more, and he mentioned that he maybe shouldn’t be going to this event at Slim’s, because he had homework to do in probability and statistics. I told him I knew something about that stuff, and he showed me his homework.

One of the problems he hadn’t been able to solve concerned 25 letters and the 25 differently addressed envelopes that they were supposed to go into. The letters are shuffled and randomly stuffed into the envelopes. What are the odds that no letters at all wind up in their proper envelopes?

I told him I thought this was an awfully hard problem to be giving to an introductory class, but we weren’t going anywhere anyway, so we got out pen and paper and we tackled it. We (mostly me) came up with a plausible looking formula for the solution, but when I calculated it, it all reduced to (24/25)25, and I knew that that couldn’t be the correct answer. (That would be the answer if each pairing of a letter to an envelope was a completely independent event from the others; but they aren’t, because each pairing takes a letter and an envelope out of the game, and that I knew that that just had to affect the odds in some way, even though I couldn’t figure out how to figure out exactly how it did. I give myself partial credit, though, for at least recognizing that our answer could not possibly be right.)

When I got back home I did some online research and discovered that there’s a special name for a permutation (that is, a shuffling of ordered objects) that does not leave any object in its correct place in the order: a derangement. It seems inconceivable that I haven’t come across this term before at some point, but I don’t have any memory of it; maybe something I encountered in college and not since.

There’s a very tidy little formula for getting the number of possible derangements of n objects: dn = n!/e rounded to the nearest integer. And therefore the probability that a random permutation will be a derangement (that every letter ends up in the wrong envelope) is the number of derangements divided by the number of permutations, or dn/n!. This is very close (just a rounding error away) to (n!/e)/n!, which cancels out neatly to 1/e, which is about 0.368 (or 36.8%), and in fact as n increases, the probability converges toward precisely 1/e very, very rapidly.

It’s neat and counterintuitive, then, to consider that the odds don’t change significantly whether you’ve got 25 envelopes to stuff or 25 trillion.

Thing is, I got all this off a website, and the calculations needed to get that simple-looking formula for dn are pretty darned long and involved. So I have to wonder what it’s doing on the homework for an introductory class. Maybe it was meant as an exercise in Internet research?

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One More Cat to Herd

Everybody’s talking like Sen. Arlen Specter’s switch to Democrat is going to make things so-o-o-o much easier for the Dems because now they have the magic number 60. That might be true if the Dems were good at the marching in lockstep thing, but they aren’t so much. Seems to me that all this means is we have one more cat to herd.

Two by Fosse

In the last couple of weeks Dave and I have watched two movies directed by Bob Fosse: Sweet Charity and All That Jazz. Dave had never seen either one, and I hadn’t seen either of them in years. I remembered not liking Sweet Charity much, but liking All That Jazz a lot. Turned out that Jazz was every bit as good as I remembered it, but Charity was a nice surprise, not a great movie musical but better than I remembered it.

The stage version of Sweet Charity seems to me to be one of the most extreme examples ever of a common problem with musicals, in which the book and the songs don’t feel like they go together. I think most of the score to Sweet Charity is terrific, and some of it is brilliant, sketching the characters with all kinds of shrewdness and compassion and humor. There are the hits, of course — I think “Big Spender” and “If They Could See Me Now” are wonderful songs — but some of the less well-known songs are great, too: “Charity’s Soliloquy” (maybe my favorite number in the show) and “There’s Gotta Be Something Better Than This” and “Baby, Dream Your Dream”.

But the book is heartless. Over and over again it sacrifices feeling, character, story, everything that might make you care about these people and their situations, for the sake of a cheap quip or easy sight gag. It’s feels as though Neil Simon thought he was supposed to be writing a sendup of the Fellini movie that the musical is based on, while the songwriters were taking it seriously.

The movie, though, changes a lot of that. Most of the jokes from the stage musical that make me cringe the most are either gone or toned way, way down and tossed off lightly so that they don’t derail the story. (Peter Stone wrote the screenplay.) The very ending of the story in the movie is different, and much better, much more meaningful than the painfully stupid gag that the stage version ends with. The ending is also clearly inspired by the ending of Fellini’s Nights of Cabiria, and in fact all the way through I got the feeling that Fosse and Stone must have been trying to capture more of the feel of that movie — the compassion, the humor (as opposed to jokes), the moments of wisdom — in this one.

Unfortunately a few of my favorite numbers from the stage version are gone from the movie — including, alas, both “Charity’s Soliloquy” and “Baby, Dream Your Dream”, though “There’s Gotta Be Something Better Than This” is here, and wonderfully choreographed. (Dave pointed out to me that in this number Shirley MacLaine lags just slightly behind the other two dancers and is always just a little bit out of line, while the other two are themselves dancing in precision, as though she were not quite fully confident in her steps and was following what they were doing — which fits the situation perfectly, as the point of the song is that she’s slowly picking up an idea from them. Fosse must have deliberately instructed MacLaine to dance it this way because she shows in other numbers that she is perfectly capable of precision with the other dancers. Sweet bit of detail.)

I thought the use of still photographs interrupting the action here and there was a nice offbeat touch, and sort of endearing, but Dave didn’t like it at all.

Given what he did in the movie, it seems like Fosse must have not liked the persistent jokiness of the stage book. Which makes me wonder: In All That Jazz there’s a brilliant scene in which the main character, Joe Gideon — obviously based on Fosse himself — is at the first reading of a new musical he’s directing and choreographing, in which his ex-wife — I’m forgetting her name in the movie but she’s just as obviously based on Gwen Verdon — will be starring. The book is filled with corny, stupid gags, and everybody in the cast is laughing uproariously throughout, except for Gideon and Verdon who clearly hate the jokes. (What’s brilliant about the scene is how Fosse conveys the sense of Gideon’s embarrassment at the material, at being in the position of directing it, and most of all at being in this position in front of Verdon, who is the only person in the room whose opinion he really respects, and what interminable suffering it is for Gideon to sit through the reading.)

Jazz makes no secret of being based in many ways on Fosse’s own life, and the musical that Gideon is working on in the movie is full of visual and verbal references to actual musicals that Fosse worked on, so I had also figured that this scene must have been inspired by some actual rehearsal or rehearsals Fosse had cringed through. So it makes me wonder now if he was thinking of his experiences directing the stage version of Sweet Charity.

The Conductor Ascending

Terrific San Francisco symphony concert tonight. Everything full of life and crisply played, every piece a highlight in its own way.

Yan Pascal Tortelier is a very bouncy conductor, both figuratively and literally, actually jumping into the air at the big moments.

The concert stated with a selection from Bizet’s L’Arlesienne music, but it wasn’t either of the suites I’ve heard before. It had an unusually large orchestra, probably because they were going to need all those musicians for the symphony at the end of the program and might as well use them for something. Delicious.

The rest of the program was pieces I hadn’t heard before. Next was Poulenc’s Organ Concerto, which is a very odd but likeable work, often changing moods in a moment. After intermission was Vaughan Williams’s The Lark Ascending, with a really stunning violin solo by Nadya Tichman.

Finishing the concert was Vaughan Williams’s Fourth Symphony, a dense, harsh, many-layered work that Tortelier nevertheless conducted without a score. I liked it a lot but it’ll take another few hearings before I really grasp it. There’s a remarkable — maybe insane is a better word for it — fugal section at the end that is a bit like watchng a tornado sweep up all the music up to that point, swirl it up, and drop it In a huge heap.

In High School, Our Head Cheerleader Used to Brag That She Was Too Pretty to Understand Math

I often like Leonard Pitts, but not his column on the tenth anniversary of the Columbine shootings. His point, essentially, is that there is no point at all in trying to understand why people do things like this; it’s just evil and he’s not evil and that’s the end of it.

Did Adolf Hitler murder six million Jews because he had a strained relationship with his father? Would it matter if he did?

As a matter of fact, the answers are, first, yes, if you can call brutal physical abuse “a strained relationship”, and second, only if you think it’s more important to try to keep other children from growing up to do the same kind of thing that Hitler did than it is for us to maintain our comfortable illusions of ego and separateness by pointing at things that grieve us and saying, I am somebody who could never possibly have been that — while never looking too closely at anything that risks upsetting our fine high notions of who we are, and aren’t.

As for him, Mr. Pitts says,

I will give them not an hour of my one and only life trying to comprehend their incomprehensible deed.

Well, yay him. Deliberately choosing not to work to understand something difficult and unpleasant — oh my yes, that’s certainly something to boast about.