Another counterexample came to me at 3 a.m. Suppose I buy a ticket in a lottery in which my odds of winning are 1 in 3. I buy a second ticket and double my odds.


You only double your odds if there were only ever 3 tickets for sale. Then you buy one & some other person(s) buy the other two.
If you buy two, and some other person buys the remaining one then your chances have doubled from 1/3 to 2/3.

Normally in a lottery doubling your entries increases your chance, but does not double it.

If you, I & Gnu bought one ticket each, and that was all the tickets, you'd have a 1/3 chance.
If Gnu & I bought one ticket each, and you bought 2, and that was all the tickets you'd have a 2/4 (1/2) chance, NOT double the previous chance.