Probability with changing number of marbles Given a bag containing 20 marbles of 5 different colors in this configuration: 
8x Blue
6x Red
3x Green
2x White
1x Black
How would you determine the probability of picking a marble of a specific color given these rules:


*

*4 marbles are picked one after the other from the bag

*Once a marble is picked, all remaining marbles of the same color are removed from the bag


So as an example of how the process might happen:


*

*From the initial 20 marbles, the first one that is picked is blue

*All the remaining blue marbles are removed, leaving 12 marbles

*The next marble that is picked is black

*There aren't any black marbles left, so 11 marbles remain

*The next marble that is picked is red

*All the remaining red marbles are removed, leaving 5 marbles remaining

*The last marble that is picked is white


So a blue, black, red and white marble was picked, and only the green marbles remained untouched.
In essence the question is given the initial configuration and the rules, what are the probabilities for each of the colors of being picked. So the probability of 1 of the 4 marbles being Blue is x and the probability of 1 of the 4 marbles being Red is y.
 A: This can be done with the inclusion-exclusion principle.  In order for a particular color marble to be drawn, it must be drawn before at least one of the other colors.  So first we sum the 4 probabilities that it occurs before each one.  Then we subtract the sum of the ${4 \choose 2}$ probabilities that it occurs before each pair of colors.  Then we add back the sum of the ${4 \choose 3}$ probabilities that it occurs before each combination of 3 colors.  Finally, we subtract the probability that it occurs before all 4 of the other colors.  The probability that a color is drawn before a set of other colors is simply the ratio of the number of marbles of that color to the number of marbles in the set of colors.
Let $N_1,N_2,N_3,N_4,N_5$ be the number of marbles for colors  $C_1,C_2,C_3,C_4,C_5$ to be selected.  For example, the probability that $C_1$ is selected is
$P(C_1) = N_1/(N_1+N_2) + N_1/(N_1+N_3) + N_1/(N_1+N_4) + N_1/(N_1+N_5) -
$$[N_1/(N_1+N_2+N_3) + N_1/(N_1+N_2+N_4) + N_1/(N_1+N_2+N_5) + N_1/(N_1+N_3+N_4) + N_1/(N_1+N_3+N_5) + N_1/(N_1+N_4+N_5)] +$
$N_1/(N_1+N_3+N_4+N_5) + N_1/(N_1+N_2+N_4+N_5) + N_1/(N_1+N_2+N_3+N_5) + $$N_1/(N_1+N_2+N_3+N_4) -$
$N_1/(N_1+N_2+N_3+N_4+N_5)$
These are the results I got by evaluating the above formula for blue, red, green, white, black respectively.

[1] 0.981858492384808 0.964382547277284 0.864940632123295 0.751062619870669
[5] 0.437755708343944

A: Running a simulation with 100,000,000 epochs shows that the probabilities should be close to:
BLUE:  0.98187599
RED:   0.9644007
GREEN: 0.86490873
WHITE: 0.75109474
BLACK: 0.43771984

Source available on GitHub
