A lottery game.

Question

Circular track - $A,B,C,D,E,A$

There are 3 such circular track. Three robots $X,Y$ and $Z$ starts from $A$ on their respective tracks. Whenever a button is pressed the robots can move forward or stay at the same place.

For $X$: probability of going $A$ to $B$ is $x_1$, $B$ to $C$ is $x_2$, $C$ to $D$ is $x_3$, $D$ to $E$ is $x_4$ and $E$ to $A$ is $x_5$.

For $Y$: probability of going $A$ to $B$ is $y_1$, $B$ to $C$ is $y_2$, $C$ to $D$ is $y_3$, $D$ to $E$ is $y_4$ and $E$ to $A$ is $y_5$.

For $Z$: probability of going $A$ to $B$ is $z_1$, $B$ to $C$ is $z_2$, $C$ to $D$ is $z_3$, $D$ to $E$ is $z_4$ and $E$ to $A$ is $z_5$.

Given $x_is,y_is$ ,$z_is$ and number of button presses, How can we calculate the probabilty of the states of the robots? For example the probability of BCA after 100 button presses.

The button is continuously pressed and the player can't see the robots. The player will write down the state on a paper and if it matches the states of the robots, he wins the lottery!!

Answer 1

Yes. Just write down the transition matrix $P$ such that

$$p_{12}=x_1, p_{23}=x_2, p_{34} = x_3, p_{45}=x_4, p_{51}=x_5,$$

and all other elements are zero. (We let $1$ represent A, $2$ represent B, etc.)

Let the number of button presses be $n$. Let the initial distribution $v = (1,0,0,0,0)$. Then,

$$\mathbb{P}(\text{at state }i) = vP^n.$$

You can find out similar probabilities for the second and third tracks. The tracks are independent of each other.

Answer 2

First u calculate the probability that it reaches B and doesn’t move next that will be $x_1$.$(1-x_2)$^99 and similarly for it comes to B and stays there for second time then probability is $x_1$$x_2$$x_3$$x_4$$x_5$$x_1$ $(1-x_2)$^94 and similarly u calculate for 3rd 4th 5th and so on because as an example u gave a large no. To me