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We had the following Markov chain:

enter image description here

I cannot see the following statement:

Starting in 0, the probability of hitting 6 is $1/4$.

I do not see because what does this mean "hitting 6"? In how many steps?

Maybe you can explain.

With greetings

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    In any. If it goes to the left it can never hit 6. It can go to right on the first try, or loop once then go to the right, two loops go to right etc. – M.B. Oct 14 '14 at 10:54

1 Answers1

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In infinitely many steps i suppose.

If you get to 4, you clearly get to 6, chances that this happens are:
After 1 step: 1/5
After 2 step: 1/5*1/5
After 3 step: 1/5*1/5*1/5

Do you understand now which infinite sum to look at?

Coolwater
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  • Ok, I see. $\sum_{i=1}^{\infty}\left(\frac{1}{5}\right)^i=\frac{1}{4}$. But why can I come to 6 in 1 or 2 steps? After one step (when ignoring the loop) I would be at 4, not at 6. –  Oct 14 '14 at 10:59
  • In one step you can come to 4, and then chances getting to 6 are 100% – Coolwater Oct 14 '14 at 11:00
  • Wouldn't it be better to say: After 1 step: 0, after two steps: 0, after 3 steps: 1/5, after 4 steps: 1/25 and so on and then to sum this which gives 1/4? –  Oct 14 '14 at 11:06
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    @math12 If you think it makes things more clear, you should do it – Coolwater Oct 14 '14 at 11:08
  • Ok, for me this seems to be indeed more clear, because in 1 or two steps it is not possible to get to 6. So in my opinion it is a little bit inprecise (at least for beginners as me) to write that the possibility to get from 0 to 6 in 1 step is $1/5$. –  Oct 14 '14 at 11:10