Find the probability of encounter between two flies on the plane (similar to gambler's ruin problem).












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Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.



I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.



My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?



Any help will be much appreciated.










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  • 1




    Try to write down an equation for the distance between the two flies...
    – Fabian
    Nov 24 at 12:34
















0














Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.



I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.



My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?



Any help will be much appreciated.










share|cite|improve this question


















  • 1




    Try to write down an equation for the distance between the two flies...
    – Fabian
    Nov 24 at 12:34














0












0








0







Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.



I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.



My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?



Any help will be much appreciated.










share|cite|improve this question













Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.



I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.



My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?



Any help will be much appreciated.







probability-theory stochastic-processes recurrence-relations






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asked Nov 24 at 12:11









MacAbra

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  • 1




    Try to write down an equation for the distance between the two flies...
    – Fabian
    Nov 24 at 12:34














  • 1




    Try to write down an equation for the distance between the two flies...
    – Fabian
    Nov 24 at 12:34








1




1




Try to write down an equation for the distance between the two flies...
– Fabian
Nov 24 at 12:34




Try to write down an equation for the distance between the two flies...
– Fabian
Nov 24 at 12:34










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Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?



Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.






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    Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?



    Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.






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      Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?



      Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.






      share|cite|improve this answer
























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        Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?



        Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.






        share|cite|improve this answer












        Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?



        Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.







        share|cite|improve this answer












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        share|cite|improve this answer










        answered Nov 24 at 17:12









        Tki Deneb

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