hello. i'm trying to optimize job performance in a warehouse. the problem is somewhat simplified here, but it gives all the necessary into.

i have 100 bins + 1 giant barrel. the 100 bins each contain 24 red balls + 3 black balls.

of the total 300 black balls, 16 have green dots, 3 have yellow dots, and 1 has a white dot. As a rule, no bin necessarily has a dotted black ball, however no bin will contain more than 1 dotted ball -- in other words each bin will contain 2 black balls with no dots and each bin will contain a 3rd black ball either with or without a dot.

in the giant barrel there are an even distribution of 2700 white balls, each numbered 1-100 (there are 27 balls with the number 1, 27 balls with the number 2, etc).

My task is to randomly select (and discard) a numbered ball from the giant barrel, then choose a random ball from whichever bin is marked on the numbered ball. the random ball from the bin is also discarded.

If i select a red ball, i try again. if i select the black ball with the white dot, i win.

Problem 1:

what is the probability at any given time that i will draw the winning ball?

Problem 2:

as i progress, i can retain information about which type of balls i have drawn from the various bins. So if i could choose the numbered balls from the giant bin, how should i calculate it?

Problem 3:

there is $1 million at stake. It costs me $1000 for each ball i take from the giant bin. I will be awarded $500 thousand for drawing the ball with the white dot, $100 thousand for each ball with a yellow dot, and $12,500 for each ball with a black dot. what is the optimum number of balls to draw (I must prepay the fee before drawing balls)?

Any ideas would be greatly appreciated.

- Jack in TO