# The Expectation of a Majority random process

##### New Member
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
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.