You have heard of the exception that proves the rule. Professor F. O. Savarese was it.

The accepted rule is that an Italian is dark and, if not actually a runt, at least not tall; that a professor is dry and pedantic, with eye trouble; and that a mathematician really lives in the strato- sphere and is here just visiting relatives. Well, Savarese was an Italian-American professor of mathematics, but he was big and blond and buoyant, two inches taller than me, and he came breezing in like a March morning wind.

He spent the first twenty minutes telling Wolfe and me how fascinating and practical it would be to work out a set of mathematical formulas that could be used in the detective business. His favourite branch of mathematics, he said, was the one that dealt with the objective numerical measurement of probability.

Very well. What was any detective work, any kind at all, but, the objective measurement of probability? All he proposed to do was to add the word numerical, not as a substitute or replacement, but as an ally and reinforcement.

“I'll show you what I mean,” he offered. “May I have a paper and pencil?”

He had bounded over to me before I could even uncross my legs, took the pad and pencil I handed him, and bounded back to the red leather chair. When the pencil had jitterbugged on the pad for half a minute he tore off the top sheet and slid it across the desk to Wolfe, then went to work on the next sheet and in a moment tore that off and leaped to me with it.

“You should each have one,” he said, “so you can follow me.”

I wouldn't try to pretend I could put it down from memory, but I still have both those sheets, in the folder marked ORCHARD, and this is what is on them:

I м ж X XІ ц ь u = - н I - Ѕ k з - - ј - ч э e - Ѕ XІ¤DІ V2pD о и D DІ ш ю “That,” Savarese said, his whole face smiling with eager interest and friendliness and desire to help, “is the second approximation of the normal law of error, sometimes called the generalized law of error. Let's apply it to the simplest kind of detective problem, say the question which one of three servants in a house stole a diamond ring from a locked drawer. I should explain that X is the deviation from the mean, D is the standard deviation, kis-”

“Please!” Wolfe had to make it next door to a bellow, and did. “What are you trying to do, change the subject?”

“No.” Savarese looked surprised and a little hurt. “Am I? What was the subject?”

“The death of Mr Cyril Orchard and your connection with it.”

“Oh. Of course.” He smiled apologetically and spread his hands, palms up.

“Perhaps later? It is one of my favourite ideas, the application of the mathematical laws of probability and error to detective problems, and a chance to discuss it with you is a golden opportunity.”

“Another time. Meanwhile'-Wolfe tapped the generalized law of error with a finger tip-”I'll keep this. Which one of the people at that broadcast placed that glass and bottle in front of Mr Orchard?”

“I don't know. I'm going to find it very interesting to compare your handling of me with the way the police did it. What you're trying to do, of course, is to proceed from probability toward certainty, as close as you can get. Say you start, as you see it› with one chance in five that I poisoned Orchard. Assuming that you have no subjective bias, your purpose is to move as rapidly as possible from that position, and you don't care which direction. Anything I say or do will move you one way or the other. If one way, the one-in-five will become one-in-four, one-in-three, and so on until it becomes one-in-one and a minute fraction, which will be close enough to affirmative certainty so that you will say you know that I killed Orchard. If it goes the other way, your one-in-five will become one-in-ten, one-in-one-hundred, one-in-one-thousand; and when it gets to one-in-ten-billion you will be close enough to negative certainty so that you will say you know that I did not kill Orchard. There is a formula-”

“No doubt.” Wolfe was controlling himself very well. “If you want to compare me with the police you'll have to let me get a word in now and then. Had you ever seen Mr Orchard before the day of the broadcast?”

“Oh, yes, six times. The first time was thirteen months earlier, in February 1947. You're going to find me remarkably exact, since the police have had me over all this, back and forth. I might as well give you everything I can that will move you toward affirmative certainty, since subjectively you would prefer that direction. Shall I do that?”

“By all means.”

I thought that would appeal to you. As a mathematician I have always been interested in the application of the calculation of probabilities to the various forms of gambling. The genesis of normal distributions-”

“Not now,”Wolfe said sharply.

“Oh-of course not. There are reasons why it is exceptionally difficult to calculate probabilities in the case of horse races, and yet people bet hundreds of millions of dollars on them. A little over a year ago, studying the possibilities of some formulas, I decided to look at some tip sheets, and subscribed to three. One of them was the Track Almanac, published by Cyril Orchard. Asked by the police why I chose that one, I could only say that I didn't know. I forget. That is suspicious, for them and you; for me, it is simply a fact that I don't remember. One day in February last year a daily double featured by Orchard came through, and I went to see him. He had some intelligence, and if he had been interested in the mathematical problems involved I could have made good use of him, but he wasn't. In spite of that I saw him occasionally, and he once spent a weekend with me at the home of a friend in New Jersey. Altogether, previous to that broadcast, I had seen him, been with him, six times. That's suspicious, isn't it?”

“Moderately,” Wolfe conceded.

Savarese nodded. “I’m glad to see you keep as objective as possible. But what about this? When I learned that a popular radio programme on a national network had asked for opinions on the advisability of having a horse race tipster as a guest, I wrote a letter strongly urging it, asked for the privilege of being myself the second guest on the programme, and suggested that Cyril Orchard should be the tipster invited.” Savarese smiled all over, beaming. “What about your one-in-five now?”

Wolfe grunted. “I didn't take that position. You assumed it for me. I suppose the police have that letter you wrote?”

“No, they haven't. No one has it. It seems that Miss Fraser's staff doesn't keep correspondence more than two or three weeks, and my letter has presumably been destroyed. If I had known that in time I might have been less candid in describing the letter's contents to the police, but on the other hand I might not have been. Obviously my treatment of that problem had an effect on my calculations of the probability of my being arrested for murder. But for a free decision I would have had to know, first, that the letter had been destroyed, and, second, that the memories of Miss Fraser's staff were vague about its contents. I learned both of those facts too late.”

Wolfe stirred in his chair. “What else on the road to affirmative certainty?”

“Let's see.” Savarese considered. “I think that's all, unless we go into observation of distributions, and that should be left for a secondary formula.

For instance, my character, a study of which, a posteriori, would show it to be probable that I would commit murder for the sake of a sound but revolutionary formula. One detail of that would be my personal finances. My salary as an assistant professor is barely enough to live on endurably, but I paid ten dollars a week for that Track Almanac.”