[FOM] throwing darts at natural numbers

[William S Heck]

On Thursday, July 30, Tom Dunion wrote:
>
> On Sunday, July 26, Arnon Avron wrote:
>
>>What will happen if our two heros through darts at the natrural numbers
>>instead of the reals? It seems to me that by the same "argument"
>>we get that nothing could possoibly happen.
>
> The answer to the question is that A beats B,  50% of the time.
>
> The reason the previous argument (giving zero likelihood either player
> can win) does not go through is that it is impossible to assign a
> probability measure on the natural numbers without some being more
> likely to be "hit" than others.
>
> Tom Dunion

Here's a game: the players throw darts at the open interval (0, 2);
the first player to hit the point at 1 wins.  Each time they throw,
they move half the distance to the board and throw at the open
interval half as wide but still centered at 1.  Does either ever win?
How about if they somehow throw an uncountable number of times?

Willie Heck