So I am having trouble drawing a markov chain and calculating the probabilities of shifting states to explain the movement of a unique "world" I am messing with.
Let me explain how the world works.
So we have a ball that launches and create a sort of ark.
The ball can travel 100 ft in the air and can move 100ft to the left so the arc is created by these movements.
There are 3 absorbing states in the chain.
One labeled X - this state is reached when the ball creates an arc without going 100ft high or 100ft to the left.
One labeled F1 - this state is reached if the ball goes higher than 100ft, imagine when this happens the ball is destroyed.
One labeled F2 this state is reached if the ball goes to the left more than 100ft, again imagine when this happens the ball is destroyed.
Those are the 3 absorbing states.
Now we have 9 other states and they are based on Standard deviations for the x and y axis.
For simplicity terms the standard deviations for each axis are at their respective %s. So 1sd is 68 feet up and 68 feet to the left, etc for 2sd and 3sd.
Each zone represents a state.
The vertical 3 states are 1 (1SD: 1ft-68ft), 2(2SD: 69ft-95ft) & 3(3SD: 96ft-99.8ft). The horizontal states are A (1SD: 1ft-68ft), B(2SD: 69ft-95ft) & C(3SD: 96ft-99.8ft).
So the other 9 states are: 1A 1B 1C 2A 2B 2C 3A 3B 3C
Every ball starts in state 1A and can move to all the states eventually reaching one of the absorbing states.
Once a ball moves on the vertical axis from state 1 to 2, it can go to from 2 to 3, or, F1 or X.
Once a ball moves on the horizontal axis from state A to B, it can go from B to C, or F2 or X.
So now, how would I draw this markov chain and calculate the probabilities of each state.
For example what's the probabilty of a ball is say state 1A to move to 1B or 2A?
I hope this is clear enough and thanks in advanced to anyone who helps!