fivemack (fivemack) wrote,

Balancing a herd of elephants

Suppose that you are in charge of the tightrope crossing a great ravine somewhere excitingly elephant-haunted.

Off the tightrope is hung a pair of rather large platforms, capable of holding eight elephants each; it is obviously vitally important that the weights placed on the two platforms balance perfectly, before sliding the platforms down the tightrope and sending the elephants to the other side. The weights of the platforms have, naturally, been adjusted to be exactly equal. Your job therefore generally involves the stewardship of an enormous number of bags of sand to be used as counterweights.

Elephants are of normally-distributed mass with a mean of four tonnes and a standard deviation of half a tonne; they come in herds consisting of sixteen independently identically distributed elephants.

You are paid ten pounds for every herd of elephants that successfully crosses the ravine, and must pay a thousand pounds to any herder whose elephants you cannot balance. One day you discover that some miscreant has thrown almost all your sand into the ravine, and you have only two kilograms of counterweight left. Can you still make a profit on average?

To generalise, what is the probability distribution of min_{A \subset \{1..2n\} : |A|=n} |\sum_{i \in A} x_i - \sum_{i \not \in A} x_i|

As far as I can see from simulation, it gets very concentrated near zero very quickly; the 99th-percentile goes down roughly as 1/binom(2n,n), whilst I was expecting it to go down as about the square root of that.
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