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    An email is sent on the network in which the recipients (0,1,2,3,4,5} are in communication.
    1 can send to 4 and 2
    2 to 1,3,5
    3 to 0,2,5
    4 to 1, 5
    5 to 0,2,4
    0 to 3 and 5
    If a message is sent to 2,3,4,5 it is forwarded randomly to a neighbour (even if this means a repeat). 0 and 1 never forward messages.
    Let ek be the expected number of time that a message starting at k is passed on.
    Find e4.

    So:

    E(X) = \sum E(X|A)P(A)

    I think I need to partition this but I'm unsure on the partition.

    Let X be the number of times a message is sent on
    E(X) = E(X|1st move is to 0)P(1st move is to ) + E(X|1st move is to 1)P(1st move is to 1) + E(X|1st move is to 2)P(1st move is to 2) + E(X|1st move is to 3)P(1st move is to 3) + E(X|1st move is to 4)P(1st move is to 4) + E(X|1st move is to 5)P(1st move is to 5)

    I'm not sure how I'd work these out using a general k to start at.

    If I assume I start at 4, as I'm trying to find e4 then I get
    e4 = 0 + 1x(1/2) + 0 + 0 + e5 (1/2)
    2e4 = 1 + e5

    e5 = 1x(1/2) + 0 + e2 (1/3) + 0 + e4 (1/3)

    I don't really think I'm going about this the right way, I would have thought I need to find a formula for starting at a general k but I don't know how.
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    Are 5 and 0 considered neighbours? This will make a difference to the answer and isn't clear to me from the wording of the question.

    My first impression is that simply working out e_4 is going to be easier than trying to find e_k.

    For e_4 we have e.g.
    P(0 times forwarded)=1/2 (i.e. 4 is to forward to the neighbour 3, but 4 cannot send to 3)

    However if 4 is to forward to neighbour 5 (probability 1/2) I need to know if 0 and 5 are considered neighbours or not to continue.
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    (Original post by Kate2010)
    I don't really think I'm going about this the right way, I would have thought I need to find a formula for starting at a general k but I don't know how.
    What you're doing looks fine. You can't really expect to find a formula for 'general' k (since each state is somewhat unique in terms of which neighbours it has), but if you carry on, you'll find you get 4 linear equations relating e_2, e_3, e_4, e_5. All you have to do is solve them, which will be a bit tedious, but not too bad I think.
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    (Original post by nota bene)
    Are 5 and 0 considered neighbours? This will make a difference to the answer and isn't clear to me from the wording of the question.
    Sorry nota bene, I don't think I explained it very well. My question gives a diagram and I tried to explain it in words but the diagram goes a bit like this:

    1----2---3
    / --- / -- /
    4---5---0

    where neighbours are considered to be those connected by an edge.
    (Ignore the dashes in the middle line, I couldn't get it to space out otherwise).


    (Original post by DFranklin)
    What you're doing looks fine. You can't really expect to find a formula for 'general' k (since each state is somewhat unique in terms of which neighbours it has), but if you carry on, you'll find you get 4 linear equations relating e_2, e_3, e_4, e_5. All you have to do is solve them, which will be a bit tedious, but not too bad I think.

    Setting up a system of simultaneous equations I got e4 = 2/3 so I think I have either set it up wrong or made a slip somewhere with the calculations.

    These were my equations:

    e0 = 0
    e1 = 0
    e2 = 1 + (1/3)(e1+e3+35)
    e3 = 1 +(1/2)(e0+e2)
    e4 = 1 + (1/2)(e1+e5)
    e5= 1 + (1/3)(e0+e2+e4)
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    I certainly don't get e4 = 2/3, so I suspect you've made a calculation error. (As a matter of course you should substitute your values for e1,e2,e3,e4 into your equations to check).
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    (Original post by DFranklin)
    I certainly don't get e4 = 2/3, so I suspect you've made a calculation error. (As a matter of course you should substitute your values for e1,e2,e3,e4 into your equations to check).
    Ok I've tried it again and subbed my values in to check and I now get:
    e0 = e1 = 0, e2 = e5 = 8/3, e3 = e4 = 7/3
    Is it ok for it not to be a whole number?
    Or have I set up the equations incorrectly?
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    Yes it's fine for it not to be a whole number (and as far as I remember, those are the numbers I got).
 
 
 
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