Chance and choice

You don't want to know why I was in Mombasa, but I'll tell you anyway, Watson. It was all because of mathematics! That surely comes as a surprise, so I had better explain. I was hot on the trail of Professor Moriarty and he knew it. He had stolen a priceless find, a fossilised humanoid skull that had been found in the Rift Valley. His problem was to get it out of Kenya. His only exit port was Mombasa and eventually he would have to go there to get a ship and make his escape. But how was I to catch him? He knew that I knew that he was heading for Mombasa but he was too cunning to go straight there and find me waiting for him. The obvious thing for him to do was to break his journey and stay for a time in Nairobi. Of course, he'd figure that I'd figure that same way and so he'd figure that I wouldn't go to Mombasa but would stop off at Nairobi to catch him. To avoid that possibility it seemed likely that he would figure not to stop off at Nairobi but go direct to Mombasa. I am up to his tricks, however, and figured it best that I ought go on to Mombasa, too.

That's all very well, Holmes, but what if he figures out that is what you have figured out?

That is perceptive of you, Watson, and though I might take that into account and change plans, accordingly, Moriarty is cunning enough to realise that likelihood and to reverse his plans, too.

It seems to me, Holmes, that each of you is able to out-think whatever stage the other has arrived at, with neither gaining an ultimate advantage.

Exactly, Watson. Although I am usually able to catch the cleverest criminals, I must concede that Moriarty makes very few mistakes.

How, then, could you find any way to give any chance of catching so elusive a villain?

Elementary, my dear Watson, I resorted to mathematics.

To mathematics? That is astounding! Please explain.

Actually, it was quite simple. If I went to the same place as Moriarty that would be a 100% success, but if I thought to catch him by stopping off at Nairobi, and he had chosen to go to Mombasa, that would be a total failure for he would make good his escape. However, if he went to Nairobi and I to Mombasa, that would not be a complete failure since I may still have another chance to catch him. Because of that second chance of success, I ought consider going to Mombasa somewhat more frequently than to Nairobi. I needed to choose my destination without bias, so I dropped ten slips of paper into a hat, six marked Mombasa and four marked Nairobi. I shuffled the slips and drew one at random. Whatever was on that slip was to be my destination. Professor Moriarty, if he were logical, ought do the same but with the odds reversed, hence, four marked for Mombasa and six for Nairobi.

And did you catch Moriarty, Holmes, and recover the fossilised skull?

No, Watson, the mathematics indicated a slightly greater chance of failure than success, actually 52% to 48%, and that's how it worked out. My slip of paper directed me to Mombasa but as Professor Moriarty was not there he must have gone to Nairobi. Still, it was a near-run thing. So now you know how I came to be in Mombasa. And there I waited for Moriarty to make his next move, and there I waited and waited and waited . but if you don't mind waiting, I'll tell you all about it during dinner.

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