What The Hare Said To Hector

Hector: Alright, explain your game to me, Mr. Hare.

Hare: First, I will stand somewhere along the race track, but I won’t tell you where, and this wall hides me from view. Now, do you see this large contraption here?

Hector: Indeed. There is a l-meter long net suspended from a high cable that extends along the track between two poles. Attached to the pole at this end (behind the wall) is a box with a crank, a lever, an anemometer, and a graphing calculator. I surmise that the crank moves the net along the cable, and that the lever releases it.

Hare: Very astute!

Hector: And I suppose that my goal is to trap you under the net?

Hare: Without any information to guide you? No such nonsense. In fact, the game is much more subtle than that. You will indeed crank the net to a position of your choosing — let’s name the end closest to you n — and then release it. I will then reveal my position — call it h — to you, and you must wager with me on whether I am under the net.

Hector: Too easy! If n \leq h \leq n + l, then I wager that you are under the net; otherwise, that you aren’t. I can’t lose!

Hare: Ah, not so fast. You see, there are random winds along the direction of the race track in these parts, and so the net may be blown considerably off from a straight downward course.

Hector: Which, of course, I won’t be able to see, due to the wall.

Hare: Precisely.

Hector: So, we could say that the net covers not [n, n + l], but [u, u + l], where u is a n-mean Gaussian process with standard deviation \sigma.

Hare: If you like.

Hector: Hence the anemometer, from which I can estimate \sigma. I see. Thus, my task is reduced to the following: given n, l, \sigma, and h, compute the probability that u \leq h \leq u + l. From there, I can decide whether the wager has a positive expected value for me.

Hare: Well played, Hector. But there is still one outstanding consideration.

Hector: And that is?

Hare: Actually computing the probability you speak of quickly enough to make your wager! I have somewhere to be promptly after this game, and as you know, we leporids hate to be late…

And with that, the Hare races off around the wall and down the track.

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