It's not the Expected Gain you want to compute, it's the Expectation, the mean of the probability distribution. In other words, if you do the draw a hundred times, say, you will get a number of white balls each time. The number will be in the range 0..17. After you do the hundred repetitions of the draw, you plot the results on a graph. On the X axis you map the number of white balls you got, 0..17. On the Y axis you map the number of times you got each number of white balls. You plot the points and draw the line. It will be a bell shaped curve. I said it would be 9, because it will peak between 9 and 10 but closer to 9.
But you can work it out without actually doing the experiment.
6 is less likely than 7, 8, or 9. And 10. Maybe even 11.
