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.