Tuesday, July 22, 2008

Improbable Stories, Statistics, and Dreams

Today was a casual day. I played with Danny and Eric and told them stories. Eric also asked a question about the likelihood of something happening, which led to me explaining some simple concepts of probability, which led to me inventing a math story while pushing him on the swing.

Traditionally, I ask for a one-sentence prompt, and then invent a story. Eric's prompt was, "I want a story about Euclid winning a game 623 times in a row."

So I made up a story about Euclid going to the county fair and paying 50 cents to play a game wherein he tossed a penny. If it came up heads, he earned 1 cent and he could toss it again. If it came up heads again, he earned another cent, and could toss it again. I explained how the probability of him getting three heads in a row was 1/2 * 1/2 * 1/2, which Eric correctly calculated as 1/8. He then calculated that the likelihood of getting four in a row was 1/16. And then 1/32. And then he did further powers in his head, stopping when he got to 1/4000 something. I will assume his math was correct; he was going to fast for me to catch up, and unlike certain males (i.e. Jon and Eric) in the family, I don't have powers of 2 memorized.

The problem with this game (other than ethical issues of gambling) was that you had to pay 50 cents per try.

Anyway, Euclid's dad, Pi, had warned him, "Do you realize that the likelihood of your getting heads 50 times in a row is (1/2)^50 ??? Do you know how improbable that is?? You're just throwing your money away!"

Since I was pushing Eric on the swing, I didn't have access to a calculator, so I didn't even attempt a hard figure. Unlike some members of the family (i.e., Jon and Eric), I am not good at doing math in my head.

You can imagine Pi's chagrin when Euclid got heads 50 consecutive times. And then 100. Pi began spluttering, "This is...almost mathematically impossible! Do you realize how extremely unlikely this is? You're learning the wrong lessons! About gambling! And about math!"

Euclid shrugged. "It's okay, Daddy," he said, "I'm just lucky."

Pi became apoplectic. "LUCK? There's no such thing as luck! There are only statistics and probabilities! You just happened to experience an astronomically improbable event!"

(Technically, I believe that each coin toss is it's own event, but why quibble? Even a brilliant mathematician can make the occasional mistake when he's close to stroke, heart failure, and catatonia.)

Eric liked the story and giggled whenever Pi began raving. By the time Euclid finally got tails (on his 624th try), Pi was being physically restrained by security.

I did end the story with Euclid trying to repeat the feat the next day during recess and failing, thus concluding that his "lucky streak" was over and swearing off life as a professional gambler.

Later, I calculated (1/2)^623 = 1/3.48 x 10 ^ 187 = 2.87 x 10 ^ (-188).

Imagine a decimal followed by 187 zeros and then 287:

There was a .000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 287
likelihood of Euclid's feat happening.

(C3PO: Sir, the possibility of successfully navigating an asteroid field is approximately 3,720 to 1.
Han: Never tell me the odds.)

(By the way, that comes out to 2.69 x 10^(-4), or .000269, which is still orders of magnitude better than Euclid's chances, though their risk was also far worse. (No, I am NOT going to try to calculate the difference. Jon can do it if he wants.))

No wonder poor Pi was apoplectic.

This afternoon, I gave both boys a piano lesson. They both did very well. Eric is further along, but Danny was an absolute star today! I was amazed at how quickly my four-year-old was integrating rhythm, reading notes on the page, playing notes on the piano, using the correct fingers, and singing along!

Nothing teaches multi-tasking like playing the piano!

Well, except for playing the organ, where a mediocre organist keeps one eye on the conductor, one eye on the music, one eye on the manual keyboard, and one eye on the pedals. (A really good organist doesn't need to look at the pedals. I am not so skilled. A really good organist also practices, doesn't make glaring mistakes, and doesn't ever wince. Again, I am an amateur.)

The organist also alternates between several different rows of keys while dancing around playing the pedals that she'd like to see but can't because she's required to wear a skirt to church. And listens to the conductor sing, listens to the congregation sing, and waits for the time lag of sound to go out, bounce off the back wall, and then return. All of these things happen at slightly different points, of course, so she has to kind of "average" them together. (Or ignore everyone and just play by an internal metronome.)

(There's a reason why Sacrament Meeting hymns tend to slow down by the third verse.)

I'm sure there are other jobs that require more multi-tasking. Perhaps being a good commander in a battle. Being an ER physician. Being the conductor of an orchestra. And, of course, the ultimate: being a Mom.

Speaking of being a Mom, it is so much fun to teach my boys, I almost want to homeschool. Eric tracks back in next week. I'll miss him. :(

Last night I had a dream that the boys broke out with the bubonic plague. They had huge purple pustules at their lymph nodes, and I kept chasing them around the house desperately trying to smear St. Ives hypoallergenic moisturizing lotion on them. They resisted and kept running away from me, and I thought, "How can they have this much energy when they are, literally, dying of the plague???"

I know that I would have done better to alert the CDC and then acquire antibiotics, but my dreams don't make sense. They're just very vivid. I was astonished when I learned, in middle school, that most people do not dream in color.

This dream comes from reading a book Cheryl gave me: The Plague Tales. It was an interesting book and I enjoyed it, although I thought its plausibility rather disintegrated at the end. It also comes because Danny's excema has broken out again.

P. S. My thanks to Jon for double-checking my figures. :)


Carolyn said...

Yes, I do hope that you have successfully beat it into Eric's head that such reckless gambling is not a good idea. :)

As for organ...I get to start lessons again in a couple of weeks! How fun! But yes, I agree they're very hard.

Gregory said...

As I spend part of my lunch time reading old entries, this one reminds me of a side story in the Ringworld series of books. If I remember the story right, Larry Niven introduced the character of Teela Brown who was the product of an attempted population control program which required parents to 'win' a lottery in order to have a child. After a century or so of this selective reproduction, the children became progressively luckier resulting in Teela Brown who was lucky in a tangible, Infinite Improbability Drive kind of way to throw in some Douglas Adams.