Estimating proportions using resampling

[moonman239]

I just thought of this problem: Suppose Ann holds a bowling competition for Joe and Bob. She has 30 pins. She puts tabs on some pins, but hides them and tells neither Joe nor Bob how many pins have been tabbed. So Joe and Bob decide to make this a fun experiment. Joe steps up, rolls the ball and hits 9 pins, two of which are tabbed. Then Bob rolls the ball and hits 11 pins, three of which are pinned. Two of the non-tabbed pins that Bob hits have also been hit by Joe. Knowing this, how can they go about estimating how many pins Ann tabbed.

My solution: We know that altogether, they hit 18 pins, 5 of which were tabbed. Knowing this, we can be 99% confident that the actual number is 5/18, +/- approx. 19.794239%.

However, I'd still like to know if there's a way to get a better estimate than that.

 

[bpet]

The bowling is essentially drawing without replacement, i.e. from a hypergeometric distribution. With this, the 99% CI is 5 to 15.