Probability: 10th ball is blue
The following is a question I've made myself, but I need help in solving it:
Suppose there are 100 balls in a box. 20 balls are blue, 30 balls are green and 50 balls are yellow. Now we randomly pick out 10 balls out of the box (one ball after the other) and we don't put the balls back in the box.
What's the probability of the 10th ball being picked having color blue?
I tried thinking of all the possibilities of the the 10th ball being blue, divided by all of the possible combinations. I tried for several hours and couldn't figure out neither of them. Help is appreciated!
Suppose that instead of picking ten balls, you pick all 100 balls and put them into a row in the order you picked them. Every one of the $100!$ possible orders is equally likely, and $20\cdot99!$ have a blue ball in the 10th position. Therefore the probability is exactly $\frac{1}{5}$.
Interesting question.
WLOG, assume that
(1) the balls are either blue (B) or not blue (N), and
(2) there are only two steps:
(i) Step One: the first 9 balls are chosen in one go ;
(ii) Step Two: the 10th ball is chosen.
Step One:
The probability of having $i \;(0\le i\le 9)$ blue balls out of $9$ chosen balls is $$\binom {20}i\binom {80}{9-i}\bigg/\binom {100}9$$. This leaves $20-i$ blue balls left, and $91$ balls left in total.
Step Two:
The probability of choosing a blue ball as the 10th ball is $\frac{20-i}{91}$.
In combination, the probability of choosing a blue ball as the 10th ball is $$\sum_{i=0}^9 \binom {20}i\binom {80}{9-i}\frac{20-i}{91}\bigg/\binom{100}9=0.2\qquad\blacksquare$$
EDIT: Just changed the lower limit of the summation from $1$ to $0$ and the result is $0.2$ (!).
Since you are (supposedly) treating all (remaining) balls equivalently at each draw, each ball has the same probability of being drawn at the 10th draw. Since the sum of these probabilities over all balls is $100\%$ (you are certain to draw exactly one of the balls as 10th) the probability for each individual ball must be exactly $100\%\div100=1\%$. Since there are $20$ blue balls, the probability that the ball drawn as 10th is one of those is $20\times1\%=20\%$.
This same symmetry argument applies regardless of the complications of the selection procedure used, as long as no information about its results are provided (if we were told the first 9 balls drawn were all blue that would certainly change the odds), and as long as the procedure is fair: it does not discriminate the balls in any way (which would for instance not be the case is say we put back a drawn ball if and only if it is blue).
Number the balls and ask: what is the probability to have ball number k as the 10th ball drawn?
This probability must be the same for every ball because none of them is favored, also probabilities must add up to 1; so it is $\frac{1}{100}$. There are 20 blue balls, each of them having $\frac{1}{100}$ chance to be the 10th.
Hence $\frac{20}{100}=\frac{1}{5}$