There are two closed boxes, filled with red and white marbles. One of the boxes is 2/3 red and the other is 2/3 white.

You're allowed to sample 5 marbles from the first box, and 30 from the second. As a result, you get 4 red and 1 white marble from the first box, and 20 red and 10 white from the second. Which is more likely to be the one with majority red marbles?

Assume the two boxes contain the same number of marbles.

If each box has 30 to 59 marbles, then the 30 marble sample is 100% probable to have been drawn from the 2/3-red box. As the population of marbles in the boxes increases beyond the 59 marbles, the probability that the larger sample having been drawn from the the 2/3-red box decreases, yet, intuitively, the larger sample of 20 red of 30 marbles would seem to come from the 2/3-red box of marbles.

But, again, 20 red marbles of 30 is 2/3 while 4 red of 5 marbles is 4/5. Overtly it would seem that 2/3 is less than 4/5 and thus the smaller sample would come from the 2/3-red box of marbles.  

Though, as the population of marbles approaches infinity, the two fractional samples both approach zero, giving each sample an equal probability of having come from either box of marbles.

Yet, we know our population must be finite. We have drawn 20 red marbles from an unknown population and 4 red marbles from an equal unknown population. 

I am weak in this area of mathematics, so I could only guess that the intuitive guess is the correct answer.

 

Edited on February 19, 2008, 8:24 am

Posted by Dej Mar on 2008-02-19 08:00:04