The Visualization of Probability

January 26, 2007

Posted by Jay Livingston

I used to play New York Lotto. I knew it was like throwing my money away. You pick 6 numbers out of 59. Let’s see: 59 C6. That’s 59! / 6! * (59 - 6)!. If you can’t do the math in your head, the Lotto ticket provides the answer on the back.

For a dollar, your chance of hitting it big is one in 22 million.

As my brother put it, quoting a statistician friend of his, “Your chances of winning the lottery are not substantially increased by entering it.”
“What do you mean?” I answered at the time. “They’re increased infinitely.”
“Yes,” he agreed, “but not substantially.”
All of which says more about the concept of infinity than about the Lotto. If you buy a second ticket, your chances have gone from 1 in 22 million to 2 in 22 million, a multiple of two. But what multiple takes you from 0 in 22 million to 1 in 22 million? None, the answer is not finite.

What puzzles me is that even though I knew the odds, I continued to buy a ticket each week. I think it was a matter of visualization. When you fill out the ticket, you pick six numbers out of 59 on a simple 6 x 10 grid, and for your dollar, you get two plays. Here’s a lottery ticket I filled out for Wednesday’s drawing. I selected my numbers by going to an online random-number generator. After all, if the New York Lotto selects its numbers randomly, a random number generator should come up with the right answer. I have also circled the winning numbers.

The results don’t look too bad. I had three direct hits, and several of the other winning numbers are pretty close to one or another of my choices. The trouble is that with only fifty-nine places to put six numbers, they’re bound to be close. The little grid doesn’t convey the concept that there are 45 million different combinations of fifty-nine numbers taken six at a time.

Then one day I was looking up a number in the phone book (only a few years from now, kids hearing that phrase will ask what a phone book is. Or was.) It took me some flipping of pages and running my finger over the columns before I came to the name I wanted. But what would have been the chances of hitting it randomly, of opening the phone book, pointing to a name, and having it be exactly the one I was thinking of? Or what would be the chances of someone picking my name at random out of the phone book?

It turns out that the Manhattan phone book has about 1.2 million names (or it would if there were one name for every line). In other words, my chances of winning the lottery were about the same as someone picking my name at random from a phone book eighteen times as thick.


This photo is only eight phone books thick. Picture a book twice this size. And then imagine someone randomly turning to your page, and then from all the names on that page, picking yours.

So now when I go past the newsstand with its Lotto sign, I think of that gigantic phone book and not the little pink-and-white grid. And ever since, I haven’t bought a lottery ticket.

Now all I have to do is figure out what to do with that dollar a week I’m saving.

I Can See Clearly Now . . .

January 24, 2007

Posted by Jay Livingston

As someone with the visual intelligence of ketchup (as Dave Barry might put it), I have great admiration and envy for people who can think in pictures – graphic designers, architects, basically anybody who can draw at a level above stick figures.

In the social sciences it’s especially useful to be able to put ideas and data in visual form. In that arena, Edward Tufte is The Man, and his 1983 book, The Visual Display of Quantitative Information, is probably the seminal work in the field. I can't remember who turned me on to it, but when I started leafing through it, I was amazed.

Tufte now has a sort of blog with an “Ask ET” forum, which has, among other things a link to a flash version of Genealogy of Pop/Rock Music: The Classic Graphic by Reebee Garofalo. It ends about 1980, but you still might want to check it out to see if Garofalo's view of, say, Meatloaf's ancestors agrees with your own.

My latest find is this periodic table (pictured below) which groups the different visualizations into families. The original site (though not the copy on this page) has a flash function so that when you drag your pointer over an “element,” it pops up an example of that type of visualization. Check it out.

Tagged

January 22, 2007

Posted by Jay Livingston

The Montclair SocioBlog has been “tagged.” If you don’t know what that is — as I didn’t— it’s just like playground tag, except that when someone tags your blog, you have to post five odd or obscure facts about yourself and then tag other bloggers.

I always hated tag when I was a kid. I was no good at it. I was slow. I was one of those kids who wore “Huskies.” Funny how feelings don’t change all that much from age nine to what now passes as maturity if not dotage.

But this is a collective blog, in principle at least, and that raises questions about the ground rules. Do we put facts about our department? About its members? I figure that some combination is probably the quickest way to get to five. So here goes.

1. I already mentioned it — the Huskies thing.

2. As a department of ten, we are fluent in Spanish, Mandarin, and Turkish, can get by in French, German, Italian, Russian, and Czech, and can toss off the odd phrase in Japanese and Yiddish. Some of us also speak English.

3. One of us shares a name with the musical brains behind the Beatles and a defensive lineman for the Giants when they won the Superbowl. Another of us shares a name with successful songwriter, and every December when he walks into stores and hears “Silver Bells,” he wishes that once, just once is that asking too much, ASCAP would send the check to his address instead.

4. None of us, as far as I know, has ever been in a rock band. Or wanted to be. Sad commentary.

5. When most of us came to Montclair, the social sciences were housed in what had been a girls’ dorm. Our offices had closets. With towel racks, even though the showers in the bathrooms and been disconnected.

I'm tagging These Pretzels Are Making Me Thirsty, the blog of Montclair grad Trish Pottersmith.

Groups and Wisdom III

January 20, 2007 
Posted by Jay Livingston

James Surowiecki argues for the “wisdom of crowds.” The average of the guesses of a lot of interested people will be closer to the right answer than will the guess of the smartest individual. If you want to get the answer to something, let them all bet on it and then watch where the money pushes the market.

The “wisdom of crowds” runs smack up against another concept in betting— the “smart money” — the conventional idea that some bettors are consistently more astute, while others are “punching bags.” After all, if the crowd, the majority of bettors, were usually right, they would long ago have driven the bookmakers out of business.

Ideally of course, a sports book makes money on the “vig,” the 10% surcharge on losing bets. (When you bet on a football game, you put up $11 to win $10. The point spread supposedly makes both sides equally attractive. If the bookie has the same amount bet on each side — say $1100 on the Bears and $1100 on the Saints —he’s guaranteed to make $100, collecting $1100 from the losers but paying out only $1000 to the winners.)

Sociologist Ray D’Angelo, who has studied bookies, says that yes, it’s the vig that the bookies count on. That plus a few out-of-control gamblers. But how often do the bets on the two sides of a game balance out? And what happens if they don’t?

One thing bookmakers do to correct an imbalance in betting is to change the point spread. By watching changes in the point spread, you can often tell which team the crowd likes. For example, in last week’s Bears-Seahawks game, the original line suggested to Las Vegas casinos was Bears minus 7. But bettors loved the Bears, and the line quickly changed to 8. Even that didn’t deter Bear bettors or attract enough Seahawks money. Oddsmakers continued to move the line up to 8 ½ and even 9. In the end, the crowd was not wise. The Bears won, 27-24, but their bettors, who gave up a lot more than three points, lost.

This week it’s the Saints and the Bears (not, as I nearly typed from force of habit, the Saints and the Roughnecks). And apparently the crowd likes the Saints. They opened as three-point underdogs. But today, some books have cut the line 2 ½ or 2, and one big book (Bodog.com) is giving Saints bettors only 1½ points. One Website that allows you to see the number of bets confirms this crowd preference: twice as many people have taken the Saints.

So do we follow the crowd? Or should we be “contrarians” and bet against the crowd? The contrarian view says that the bookies stay in business by being smarter than the public. Bookmakers probably also subscribe to the smart money view. That’s why Ray D’Angelo’s small-time bookmakers didn’t worry about bets from “out of control” gamblers. Those bettors were definitely not smart money.

But some bettors really are the smart money. I once heard an interview with a man who sets the line for one of the big Las Vegas casinos. He said he might not be worried by a lot of money from the general public coming in on one side. But there are particular sports bettors whose opinion he respected so much that even a relatively small bet from one of them would cause him to move the line.

My guess is that in tomorrow’s game, it’s the sheer volume of money on the Saints, not the bets of a few experts, that has pushed the line down. In any case, if you’re a contrarian, you’ll go with the Bears (also if you’re a Chicagoan, but that’s a different matter). If you believe in the wisdom of crowds, you’ll bet the Saints.

There’s one more risk in going with the crowd when their betting has moved the line — the worst-case scenario: You call up your bookie on Sunday and find that all the money coming in on the Saints has driven down the line. Instead of getting three points, the line is 2 ½. You figure, hey, it’s only a half-point, a minor consideration far outweighed by the wisdom of the crowd. You take the Saints and settle in to watch the game. It’s a close one, tied for much of the fourth quarter, right up until the final seconds, when the Bears kick a field goal to win 24-21. If you had been able to get the three points, you'd have a push. But the crowd pushed the line down to 2½, leaving you a half-point short, and you hurl your copy of The Wisdom of Crowds through the TV screen.

UPDATE: The Bears won 39-14. The bookies cleaned up, and the crowd was left to reconsider its collective wisdom.