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Thread: The Expectation of a Majority random process

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    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.

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    Re: The Expectation of a Majority random process


    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.

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