Consider a polyurn scheme of more than two colors. Let us draw a ball from the urn and replace it with another ball of the color we picked from the urn. We assume that $w$ is the number of white balls in the total of $N$ balls.
If $S_n$ is the number of white balls in a random sample of $n$ random draws, we need to obtain a recursive equation for $\Pr(S_n=x)$, where we have $P(S_1=x)= \left(\frac{W}{N} \right)^X \left(\frac{N-W}{N} \right)^{1-X} ,$ and we have $X=0,1$.
My approach1:
Given: Sn= Number of white balls in a random sample of n draws. To start with , we have P(Sn=x) where x =1 , which means we picked a white ball, the probability of picking a white ball can be written as: W/N.
Next we have P(Sn=x) where x=0, which means we did not pick a white ball. Then the probability of NOT picking a hite ball can be written as 1-W/N.
If we combine the first 2 equations i just obtained, we get the probability p(SN=X) = (W/N)^X * (1-W/N)^(1-X) => lets call this equation 3. where u put in values of X as 0 and 1. Lets put X=1 in equation 3, we get W/N. Lets put X=0, we get (1-W/N)
My approach 2:
P(S2 = 1) which is the probability of getting one white ball in a sample of (n=2) draws. There can be two ways of drawing the said combination. White- Non White or Non white - Whit. therefore the probability can be written as:
= 2 * (W/N)*(N-W/N+1)
P(S2=0) which is the probability of getting ZERO white balls in a sample of (n=2) draws.
= 2 * (N-W/N+1) * (N-W+1/N)
P(S3 = 1) which is the probability of getting one white ball in a sample of (n=3) draws. The various forms the draws can assume are => W-NW-NW , NW-W-NW, NW-NW-W. (Where NW=> Non white and W=> White) When we write down the probabilities we obtain:
= [3*(W/N)(N-W/N+1)(N-W+1/N+2)]
P(S3=0) which is the probability of getting ZERO white balls in a sample of (n=3) draws. The various combinations possible are: NW-NW-NW during all the three draws since we get zero white balls. Writing the probabilities: = [3*(N-W)(N-W+1)(N-W+2)] / [N*(N+1)*(N+2)]
P(S4 = 1) which is the probability of getting one white ball in a sample of (n=4) draws. The various combinations possible are: W-NW-NW-NW , NW-W-NW-NW, NW-NW-W-NW, NW-NW-NW-W (Where NW => NON WHITE & W => WHITE) Writing the probabilities: =4*(W/N) * ((N-W) / (N+1)) * (N-W+1/ N+2) * ((N-W+2) / (N+3))
P(S4=0) which is the probability of getting ZERO white balls in a sample of (n=4) draws. The various combinations possible are: (NW-NW-NW-NW ) *4
=4*(N-W/N) * (N-W+1)/(N+1) * (N-W+2 / N+1) * (N-W+3 / N+3)