The Expectation of a Majority random process

I want to know the answer of the following question, I will be so thankful if somebody gives me some clues:
"We start a random process with Nb blue balls and Nr red balls such that N=Nr+Nb. In each step we divide all these N balls into groups of size 4 at random and in each group if the number of blue balls is 3 or 4, then all balls in that group change their color to blue, unless they all change to red. If we denote the number of blue balls in step t by Nb(t), then what is the expectation of Nb(t)?"
Thank you so much in advance.


TS Contributor
Deriving the exact distribution maybe difficult and tedious, but you can find out the recursive relationships between the expectations.

Now we are given that \( N_B(1) = N_B, N_R(1) = N_R = N - N_B \)

For any randomly choose group of size \( 4 \) in step \( t \), the number of blue ball follows a hypergeometric distribution. The change of number of blue ball in this group is listed as the following:

\( \begin{tabular}{|c|c|} \hline
\text{Number of blue balls in this group} & \text {Changes of number of blue balls in this step}\\ \hline
0 & 0 \\ \hline
1 & -1 \\ \hline
2 & -2 \\ \hline
3 & 1 \\ \hline
4 & 0 \\ \hline

Now you can try to compute the expected changes of number of blue balls in a particular group in step \( t \). Since expectation is linear, the expected number of change of total blue balls is equal to the sum of all these \( \frac {N} {4} \) groups (although they are dependent).

So the rest of the details can be filled by you.