In the final scene of Hitchcock’s Dial M for Murder, Tony Wendice – who had plotted the assassination of his wife Margot – returns home and tries to open the front door. As there is no key in his pocket, he uses the key in his wife’s handbag, which he just retrieved from the police station. But the key doesn’t fit. He is about to go around to the back entrance, when he remembers there is another key under the carpet in the stairway – the key he had given Swann, the hired assassin, to enter the flat. He finds the key and opens the door, thereby proving his guilt.

Margot, her lover Mark Halliday and Chief Inspector Hubbard are inside the flat. While waiting, the Inspector explains:

Mark: What happens now?
Inspector: Sooner or later he’ll come back here. As I’ve pinched his latchkey, he’ll try the one in the handbag. When that doesn’t fit, he’ll realize his mistake, put two and two together and look under the stair carpet.
Mark: But if he doesn’t do that, all this is pure guesswork. We can’t prove a thing.
Inspector: That’s perfectly true, but once he opens that door, we shall know everything.

Tony’s action is a Smoking Gun: there is no way that he would have opened the door if he hadn’t known about the hidden key. And he only knew about the key because he put it there. The probability that he opens the door if he is innocent – a False Positive – is zero: FPR=0%. Hence, if he does open the door, he is certainly guilty: PP=100%.

But what if Tony hadn’t open the door? Would that have proven his innocence? No. There is a small chance – say 5% – of a False Negative: Tony may have forgotten about the key, or may not be focusing on it, or might consider it prudent not to use it and prefer to go around to the back entrance, as indeed he was about to do. Hence, TPR=95%: the probability that Tony opens the door if he is guilty is high, but not 100%. As a consequence, if he does not open the door, he is not necessarily innocent: NP>0%.

The following figure illustrates the difference between Perfect Evidence (TPR=100%, FPR=0%) and a Smoking Gun (TPR=95%, FPR=0%).

Perfect Evidence is infallible. If it is there, the hypothesis is certainly true: PP=100%; if not, the hypothesis is certainly false: NP=0%. With Perfect Evidence, posterior probabilities are independent of the Base Rate: if Tony opens the door, he proves himself guilty; if he doesn’t, he proves himself innocent, whatever the Inspector might have suspected.

A Smoking Gun, on the other hand, is conclusive if the evidence is positive: PP=100%, but not if it is negative: NP>0%. NP is a convex function of BR, with the degree of convexity directly dependent on TPR. Before the door test, the Inspector was strongly suspicious of Tony: his BR was high. Therefore, if Tony had not opened the door, the Inspector would have updated his Base Rate downwards – the more so, the higher TPR. With a lower NP, he would not have been able to “prove a thing”. But he would have remained suspicious of Tony’s guilt.

Sometime earlier, Margot – who had killed Swann in self-defence as he tried to strangle her, but failed to convince the jury and was sentenced to death for murder – had been secretly taken home from prison. She had also tried and failed to open the door with the key in her handbag but, unlike her husband, had proceeded straight to the back entrance.

Margot is the only other suspect. So the Inspector sets her to go through the same door test:

Margot: Why did you bring me here?
Inspector: Because you were the only other person who could have left that key outside. I had to find out if you knew it was there.
Inspector: You didn’t.

If Margot had opened the door, she would have proved herself guilty, just like her husband did shortly afterwards. But she didn’t. Does that prove she is innocent? Again, no. The same small chance of a False Negative applies to her as well. Margot did what her husband could have done. So why does the Inspector take such a different stance on her? If Tony had not opened the door, the Inspector would still have been suspicious of him. Why is he not as suspicious of Margot? The answer is that he had assigned a small Base Rate to Margot’s guilt. As the figure shows, in a Smoking Gun a low BR implies a positive but low NP. To reach complete certainty, the Inspector would still have to run the door test on Tony. But, even before that, he strongly believes that Margot is innocent. Starting from different priors, the same evidence leads to different conclusions.

Here is the final twist. Before the door tests, irrespective of what the Inspector believes, in the eyes of the Court Margot had a much higher probability of guilt compared to her husband. So much so that she had been sentenced to death. When Tony opens the door and sees the three waiting for him, he feels lost and gives up. Had he kept his presence of mind, however, the movie could have had a very different ending:

Tony: Margot! You here?
Inspector: Mr. Wendice, how did you open the door?
Tony: With this key. It was exactly where Margot told me I would find it.

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