Keith Briggs

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### The lottery - an order statistics paradox?

Suppose 6 balls are drawn uniformly from 49 balls (numbered 1 to 49), without replacement.

Sort the balls by the value of their numeric label and let xi be the value of the ith ranked ball for i from 1 to 6.

Then, for k = 1,2,3,...,49:

(Proof: homework exercise.)
These distributions look like this for i = 1,2,3,4,5,6; and have the mean and mode as in the table. (Proof: another homework exercise.)

imeanmode
150/71
2100/710
3150/720
4200/730
5250/740
6300/749

In other words, the most likely value for the smallest numbered ball is 1, the most likely value for the second smallest numbered ball is 10, and so on. Everything I have claimed so far is correct.

Therefore, you should put your money on balls 1,10,20,30,40, and 49.

The last statement is nonsense. Why? (Don't tell me; I know.)