The Expectation of a Majority random process

#1
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.
 

BGM

TS Contributor
#2
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:

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

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.