L.A. Math: Romance, Crime, and Mathematics in the City of Angels

by James D. Stein

  I’ve learned to humor Pete’s forays into mathematics. After all, I’ve profited big-time from some of them. “What good does that do you, Pete?”

  “All bell-shaped curves have the same shape. There’s a 68% probability of being within one standard deviation of the mean. Since the mean is eighty and one standard deviation is four, that means that 68% of the time that Theresa shoots a hundred free throws, she’ll sink between seventy-six and eighty four. And 95% of the time, she’ll be within two standard deviations of the mean and sink between seventy-two and eighty-eight. Only about once in five hundred times will she be further away from the mean than three standard deviations. This time she’s fourteen free throws below her average, and 14 divided by 4 is 3½. So she’s three and a half standard deviations below her mean. You don’t get numbers like that by blind chance, Freddy. Something’s going on.”

  (Binomial distribution continued on p. 208)

  He took the basketball I was holding. “I’ll take care of the coaching, Freddy. Find out what’s happening, or we’re all in the soup.”

  My years of training had not been wasted. When you want to know something about someone, the best way is to ask a friend. I called Pete’s cousin Irene.

  It was a piece of cake. I herewith report our conversation in full.

  “Irene, has something happened to Theresa?”

  “Didn’t she tell you? She won’t be able to play in the game Saturday!”

  “Did something happen to her, Irene? Has she had an accident?”

  “No, she’s fine. But the most amazing thing happened. She got a phone call last evening, and now she can go see the KrystalVision concert Saturday night!”

  I read the entertainment pages, so I knew that KrystalVision was hot, especially if you were a thirteen-year-old girl. “I thought that concert was sold out, Irene.”

  “It was. But somebody said Theresa won two tickets in a rock station giveaway, and she told me I could have the other one.”

  I grasped at a straw. “But, Irene! Think of your teammates. You’ll let them down terribly.”

  “No problemo. I told them all about it and promised them all autographed pictures of KrystalVision.” She rang off.

  I resolved then and there to limit both my investments on, and my future dealings with, teenage girls—at least with girls in the thirteen- to fifteen-year age group. Pete gritted his teeth when I reported.

  “This is Ollie’s doing, Freddy. I just know it. One of his relatives is a ticket scalper. Well, it’s water under the bridge, I’m afraid. We’ll just have to do the best we can.”

  There went a hundred bucks. Basketball practice the next day was dismal. Theresa and Irene showed up, but their heads and hearts were obviously somewhere else. Thursday’s practice was every bit as depressing. Even if, by some miracle, KrystalVision canceled their performance on Saturday, Theresa would be so upset that she might cease to be an offensive force. She no longer seemed to be an offensive force, anyway, as she still couldn’t throw a pea in the ocean.

  I had mentally written off the hundred bucks and resigned myself to having to endure Ollie’s gloating, to say nothing of the loss of a C-note, when a bolt from the blue arrived Friday morning. This particular bolt took the form of a bulletin from the District School Board, a copy of which, addressed to Pete, arrived in the morning mail. Pete was wolfing down the Friday brunch special at Irv’s Deli, so I took the liberty of opening it.

  The bulletin announced that disturbing events had come to the board’s attention, and as a result it was forced to cancel the championship game on Saturday night. The board would have preferred to postpone the game, but the school term was coming to an end, and so there would be no opportunity to play the game at a later date. Rutherford and MacMillan were being declared cochampions, and the awards banquet would go on as scheduled. The last paragraph mentioned that a school district representative would go to each school in the district next week to give a lecture entitled “The Evils of Gambling.”

  Curiouser and curiouser. If nothing else, though, I felt that Pete would be somewhat relieved not to face the ignominy of losing the championship game to Ollie. Of even more importance, I could now be assured of not losing a hundred bucks. When Pete arrived back from Irv’s Deli, I handed him the announcement.

  He scanned it quickly. “I sort of expected something like this, Freddy.”

  Pete always gives me the impression that, on the day that the sun finally rises in the west, he will say, “I sort of expected something like this, Freddy.” While I was certainly relieved to have the hundred bucks back in my pocket, my curiosity was itching.

  “How on Earth could you possibly have predicted that the school district would cancel the game?”

  “I couldn’t be sure. However, you know that Arnie Schrafft was booking the game.”

  “So?”

  “You remember that the line all of a sudden went from MacMillan plus four to MacMillan minus two? Well, that had to be due to a lot of money coming in on MacMillan. Arnie’s fairly new to bookmaking, and he did the obvious thing, raising the price on MacMillan to attract more betting on Rutherford.”

  “That certainly makes sense.”

  “It makes sense, but there’s a huge risk. Arnie could get seriously middled.2 If the game ended with MacMillan either winning by one or losing by three or less, he’d have to pay off everyone who bet on MacMillan while the line on MacMillan was at plus four, as well as everyone who bet on Rutherford later on and got two points. And I’m pretty sure that a large chunk of his action came from Ollie betting on MacMillan plus four.” He paused. “When that happens to a big-league bookie, they just lay off the extra action. Arnie’s a small-timer, and where could you lay off action on a girl’s junior high basketball game? I didn’t think he could stand the pressure of having to pay almost everyone if the game ended unfavorably for him.”

  “So what do you think he did?”

  “I’m virtually certain I know what he did. He let his superiors know that there was betting going on and that one of the players had taken a bribe—even though I’m not sure that Theresa accepting concert tickets constitutes a bribe. She’s probably completely unaware of what’s been going on behind the scenes.” He looked thoughtful. “By the way, Freddy, do you suppose you could trace those tickets?”

  “What tickets?”

  “The ones Theresa got. If I could show they came from Ollie …” His voice trailed off.

  I gave it a thought. “It depends. But is it really worth it? And even if I could, it’s not a crime to send tickets to a rock concert to a minor.” I thought about it for a moment. “Although maybe it would be a better world if it were.”

  “At least, a quieter one.”

  CHAPTER 12

  IT’S ALL IN THE GAME

  “Is Señor Lennox at home?”

  I couldn’t help but feel that I recognized the slender, fortyish woman with the slight Latino accent on our doorstep, but I couldn’t immediately place her. It was one of those situations where you normally see someone in one particular location, and then when you run into him or her elsewhere, you do a double take. This particular double take took about five seconds, and then I had it.

  “Come on in, Dolores. I’ll take a look.”

  Whenever you read about Los Angeles, you read about Hollywood, or sports franchises, or riots, or gang problems, but you never read about the people like Dolores who make up Los Angeles, and that’s a pity. Twenty years ago, Dolores and her husband had just started Casa Dolores, a small Mexican restaurant on the Brentwood–Santa Monica border, when he was killed in a traffic accident, leaving Dolores to care for Maria and Pedro, their two children. In those twenty years, Dolores not only managed to make a success of the restaurant but also put aside enough to make sure that they each had a college education. Maria is doing her residency at the University of California, San Francisco, and Pedro is about halfway through USC. We’re regulars at Casa Dolores, and Pete told me that when Pedro and Maria w
ere in high school, they used to wait tables and clean up far into the night. Maria was a whiz at science courses, but Pedro had a little trouble with algebra, and Pete would tutor him in exchange for free meals.

  I had gotten Dolores settled in the front room when I was spared the problem of locating Pete, who had evidently heard the door opening. It wasn’t a double-take situation for him, possibly because Dolores had brought Pedro over evenings for tutoring. He greeted her warmly.

  “It’s nice to see you again, Dolores. I don’t have to ask you how the restaurant’s doing; it’s always packed. How are Pedro and Maria?”

  “They are both well, Señor. Maria, she will soon become a doctor!” Dolores pronounced the ‘r’ in Maria a lot more attractively than Pete.

  “I’m not surprised. Tell me, Dolores, what can we do for you?”

  She hesitated a moment. Then she took out a folded piece of paper and passed it to Pete. “I would like to consult you about what I should do about this.”

  The paper was a black-and-white copy of the front of a baseball card. It was a picture of Babe Ruth. On the front of the card were two dated autographs—one made by Babe Ruth in 1927 and the other by Roger Maris in 1961. Pete whistled.

  “If this is yours, Dolores, put it in a safe deposit box quick! If the autographs are genuine, they are dated on the days that Ruth hit his sixtieth home run and Maris his sixty-first. A collector would give a fortune for it.”

  “They are genuine, Señor. And the card is safe.” Dolores paused. “It was left to me by my grandfather. His father, my great-grandfather, used to play baseball in the Mexican leagues, and he played against Babe Ruth. They went out drinking together. My great-grandfather, he was watching the day Babe Ruth hit his home run, and he got Babe Ruth to sign the card. He was also watching when Señor Maris was playing, and he waited for three hours after the game to get him to sign the card.”

  “Then what do you need me for, Dolores?”

  “I want to sell it. You see, I have investigated how to sell a card like this. One does it through auctions. There are two places in town where I could sell this card.”

  Pete nodded. “I’ll bet I can guess. Classic Collectibles and Vintage Memories.”

  Dolores smiled. “I knew you would be helpful, Señor. Those are the two places that handle cards like this. I looked at the records of their auctions for the past ten years.”

  “You’re very thorough, Dolores.”

  She was serious. “It is a business, Señor. Like running the restaurant. You must have good suppliers at a fair price. But with suppliers, you can give them a tryout. If they give you good food at a fair price, then you give them a longer contract. Here it is different. I only have one card to sell.”

  She took out another sheet of paper. “Each auction house has what they call a reserve price. It is a minimum bid. I have talked to the owners of both places. Classic Collectibles will set a reserve price of $20,000, and Vintage Memories is willing to set one for $30,000.”

  It didn’t look like much of a problem to me. But then Dolores continued, and I saw where the problem was.

  “In looking at the auction results, Señor, I noticed that the high-priced items sell for more at Classic Collectibles. They sometimes receive prices of $100,000 for their items, but the highest I could find at Vintage Memories was about $70,000.” She paused. “I have been looking at these numbers so long I am going crazy. Some mornings I wake up feeling happy, and I know I am going to get lots of money for the card. So I feel I should auction it at Classic Collectibles. Some mornings I wake up unhappy, and I say to myself $30,000 is a lot of money. So I feel I should auction it at Vintage Memories. After going round and round for a week, I thought of you helping Pedro with algebra, and how smart Pedro says you are.”

  Pete nodded. “I think I can help you, Dolores.” He took a piece of paper and made a drawing, which I have reproduced here:

  “I think the above diagram summarizes the information you gave me,” Pete asserted. “The numbers represent how many thousand dollars you expect to get. For instance, if you auction the card at Vintage, and you get a high price, you told me that you might make as much as $70,000.”

  Dolores and I both studied the diagram. “That’s right,” she said. “But how does that tell me what to do?”

  “You should auction the card at Vintage,” Pete said, “for no matter what happens, you can assure yourself of $30,000, and if good things happen, you might make as much as $70,000.”

  “But I knew that,” she protested. “Why shouldn’t I sell it at Classic? I could make $100,000!”

  Pete thought a minute. “Here’s how to see it. Suppose that only one person wants to buy the card. If you auction it through Classic, it will only cost $20,000. If you auction it through Vintage, it will cost $30,000. If more than one person wants to buy it, then you will be happy to let them fight it out and make you more money.”

  Dolores was still looking dubious. “Here’s another way to look at it,” Pete said. “Suppose you had a lot of these cards to auction. Imagine that the buyers realize that there are enough cards to make them all happy. What would they do then?”

  Dolores thought for a bit. “I see, Señor. They would make an agreement among themselves to buy all the cards at the lowest possible price. That’s what I do when I have several competing suppliers. Now I see what to do and why I should do it.”

  She reached inside her purse and took out a checkbook. “You have given me good advice, Señor Lennox, and I will do as you suggest. What do I owe you?”

  Pete looked at me. “Do you have any plans this evening, Freddy?”

  I knew where he was heading. “None I couldn’t cancel.”

  “Then cancel them.” He turned to Dolores. “I’m just tutoring you instead of Pedro. You owe Freddy and me dinner tonight. And a bottle of sangria.”

  “You are most generous, Señor. You may have saved me $10,000. Dinner is a small price to pay.”

  Pete smiled. “We’re going to be very hungry.” We all shook hands, and she left.

  When she had gone, I took another look at the diagram that Pete had drawn. “Is it always right for her to auction the card at the place that offers the highest reserve price?” I asked.

  “In a situation like this, yes,” he replied. “But there are other situations in which the picture isn’t so clear.”

  A few weeks later, we were dining at Casa Dolores, and Dolores came over and told us she had sold the card for $62,000. She was so delighted that our dinners there are on the house until we start abusing the privilege, so I think we’ll be eating a lot of Mexican food. While I was happy with the result, I was ready to kick my parents all the way back to Altoona, Pennsylvania. Our family had moved from there when I was about twelve, and when we moved, they made me leave three shoeboxes of baseball cards behind!

  I’ve picked up a few pointers from Pete (and vice versa, I might add), and I’ve discovered that some of the things I learn from watching him have an odd way of showing up in other places. A couple of weeks later, I was confronted with a personal problem. It kept gnawing at me, and after thinking about it, I decided I might try to apply Pete’s methods to it. But I couldn’t get it to work, so I figured I must be doing something wrong. Pete and I must have been on some sort of psychic wavelength because when I opened the door, there he was. He was just about to open his mouth, but I got the first word in edgewise.

  “Pete, how’d you like to help a buddy and business partner?”

  He looked at me. “What kind of help?”

  “It’s probably right up your alley. Do you remember that analysis you did for Dolores a few days ago?”

  “Of course. Speaking of Dolores, are you hungry?”

  “Not just now. I’ve got a problem. I haven’t heard from Lisa in a while, and so I was trying to decide whether to call her. It’s like that old song: Once in a while she won’t call, but it’s all in the game. It occurred to me that I had two choices: to call or not to call. And the
re are two possibilities for her: either she wants me to call, or she doesn’t.”

  Pete thought for a second. “Well, if you want to apply game theory, you have to make up a chart like the one I did for Dolores.”

  I nodded. “Yeah, I know. I’ve given it a little thought, and here it is.” I shoved a piece of paper at him, which I’ve reproduced below. “I guess I’d describe the numbers as how happy I feel I would be on a scale of one to ten.”

  Pete looked at the diagram. “I think I see what you’re saying. Obviously, the best case is that you call and she wants to hear from you. The worst case is that you call and she doesn’t want to hear from you. Not only is it embarrassing, but it looks like your relationship is down the drain.”

  I nodded. “That’s right. And if I don’t call and she wants to hear from me, I’ve made the wrong move. Lisa feels that when a relationship is good, each of you knows how the other is feeling. Even though I’d have the wrong take in this situation, at least she wants to hear from me. That’s some consolation. Finally, if she doesn’t want to hear from me and I don’t call, at least I avoid looking like an idiot, and I can always hope that her mood will change.”

  “Okay.” Pete thought for a moment. “You’ve got a second hand on your watch, don’t you, Freddy?”

  “No, it’s digital. Why?”

  “That’s okay. Look at it, and tell me how many seconds it reads.”

  I certainly didn’t know what he had in mind, but I followed instructions. “Fourteen seconds.”

  “In that case, call her.”

  I looked at him. “Are you trying to be funny?”

  Pete shook his head emphatically. “Not a bit. Game theory dictates that you should call her one time out of three but make your decision on a random basis. If your watch had read between zero seconds and twenty seconds, as it did, I would have advised that you call her. Otherwise, I would have suggested that you don’t call.”