How Does Probability Affect Point Dynamics in a Circular Chain of People?

[ajaz706]

I am facing difficulty in solving this problem.. Anybody can help? The problem is as follows:

There is a circular 1D chain of two types of persons (say + and -). for example something like +-+--+---++++-+-+ etc. but this is circular. Now each person jumps with probability, + jumps with 'x' probability and - jumps with 'y' probability at time t. the rules of dynamics of this system are as follows.

If a person jumps before its right person, it gains a point and the right person loses that point. If the person jumps before its left person, it loses an point and the left person gains a point.

How can this system say (number of + persons) be described as an evolving function of time i.e. +(at time t+1) in terms of +(at time t).

 

[mmwave]

Hello,

I understand the difficulty you are facing in solving this problem. It is an interesting and challenging one. I would approach this problem by first breaking it down into smaller, manageable parts.

Firstly, let's define some variables to represent the number of + and - persons in the chain at any given time t. Let's use N+ to represent the number of + persons and N- to represent the number of - persons.

Next, let's consider the probability of a + person jumping at time t. According to the rules of the system, this probability is x. Similarly, the probability of a - person jumping at time t is y.

Now, let's think about the possible outcomes of a jump at time t. There are two possibilities: either the person jumps before their right neighbor or they jump before their left neighbor.

If they jump before their right neighbor, the + person gains a point and the - person loses a point. This would result in an increase of 1 in N+ and a decrease of 1 in N-. On the other hand, if they jump before their left neighbor, the + person loses a point and the - person gains a point. This would result in a decrease of 1 in N+ and an increase of 1 in N-.

Based on these outcomes, we can write the following equations:

N+(at time t+1) = N+(at time t) + x(N-(at time t) - N+(at time t))

N-(at time t+1) = N-(at time t) + y(N+(at time t) - N-(at time t))

These equations represent the change in the number of + and - persons at time t+1 in terms of their values at time t.

To get an evolving function of time for N+, we can substitute the second equation into the first one:

N+(at time t+1) = N+(at time t) + x[N-(at time t) - (N-(at time t) + y(N+(at time t) - N-(at time t))]

Simplifying this equation, we get:

N+(at time t+1) = N+(at time t) + x(N+(at time t) - N-(at time t))

This equation gives us a relationship between N+(at time t+1) and N+(at time t). We can use this equation to calculate