X-files

[radosr]

Fox Mulder is standing in the middle of a perfectly circular field. Martian zoologists are landing randomly at points on the circumference of the field. They land at one minute intervals, starting at midnight. As soon as there are Martians at points A,B,C such that triangle ABC contains the center of the field, Fox will be teleported to the waiting space-ship and spend the rest of his life as a guinea pig for Martian experiments. What is the expected time until he is abducted?

 

[pruse]

The answer is 4 minutes.

The probability that (n) randomly and uniformly chosen points on a circle do not contain the center is [imath]\frac{n}{2^{n-1}}[/imath] (a nice and simple solution can be found here Semi-circle covering n points : Puzzle). Now let (X) be the number of minutes it takes Mulder to be abducted, and define [imath]X_n[/imath] to be 1 if the first (n) points do not contain the center, 0 if they do. Then [imath]X=\sum_{n=1}^\infty X_n[/imath], so

[imath]E[X]=\sum_{n=1}^\infty E[X_n]=\sum_{n=1}^\infty\frac{n}{2^{n-1}}=4[/imath]

 

[koupparis]

I think that sums to 4, not 3.

 

[Tsotne]

I think that sums to 4, not 3.

That sum is exactly 4.

 

[pruse]

You're right. I missed a 1 when doing it. thanks.

 

[radosr]

You're right. I missed a 1 when doing it. thanks.

Very nice!