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Math Help - Inhabitants probability

  1. #1
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    Inhabitants probability

    Hello,

    I have to solve the following example:

    "In a town of n+1 inhabitants, a person tells a rumor to a second person, who
    in turn repeats it to a third person, etc. At each step the recipient of the rumor is
    chosen at random from the n people available.
    a) Find the probability that the rumor will be told r times without ever returning
    to the originator."

    The probability that the second person choose one, that is not the first one, must be (n-2)/(n+1) and also for the 3rd,4th,...,r person. So the final probability should be \frac{(n-2)^r}{(n+1)^r}, correct?
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  2. #2
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    Re: Inhabitants probability

    Hello,Curtius!

    I think you have miscounted.
    But then, maybe I did, too . . . Please check my reasoning and my work.


    \text{In a town of }n+1\text{ inhabitants, a person tells a rumor to a second person,}
    \text{who in turn repeats it to a third person, etc. }\;\text{ At each step the recipient}
    \text{of the rumor is chosen at random from the }n\text{ people available.}

    \text{a) Find the probability that the rumor will be told }r\text{ times}
    . . . \text{ without ever returning to the originator.}

    The originator [1] can tell any of the other n people.
    . . P(1) \,=\,\frac{n}{n} \,=\,1

    [2] can tell any of the other n-1 people ... excluding [1].
    . . P(2) \,=\,\frac{n-1}{n}

    [3] can tell any of the other n-1 people ... excluding [1], but including [2].
    . . P(3) \,=\,\frac{n-1}{n}

    [4] can tell any of the other n-1 people ... excluding [1], but including [2] and [3].
    . . P(4) \,=\,\frac{n-1}{n}
    . . . . . . \vdots

    [r] can tell any of the other n-1 people.
    . . P(r) \,=\,\frac{n-1}{n}



    \text{Prob} \:=\:\underbrace{1\cdot \frac{n-1}{n} \cdot\frac{n-1}{n}\cdot\frac{n-1}{n} \cdots \frac{n-1}{n}}_{r\text{ factors}} \;=\;\left(\frac{n-1}{n}\right)^{r-1}

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  3. #3
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    Re: Inhabitants probability

    Thank you very much. I have also an other question. Please check if it is right.

    The second part of the exercise is the following:

    "b) Find the probability that the rumor will be told r times without ever being
    repeated to anybody."
    P(1)=\frac{n}{n}=1

    P(2)=\frac{n-1}{n}

    P(3)=\frac{n-2}{n}

    .
    .
    .

    I am not sure what the prob. is for P(r).
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  4. #4
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    Re: Inhabitants probability

    Hello again, Curtius!

    \text{Find the probability that the rumor will be told }r\text{ times}
    . . \text{without ever being repeated to anybody.}

    . . \begin{array}{cccccc}P(1)&=&\dfrac{n}{n}&=&1 \\ \\[-3mm] P(2)&=&\dfrac{n-1}{n} \\ \\[-3mm] P(3) &=& \dfrac{n-2}{n} \\ & \vdots \end{array}

    \text{I am not sure what the prob. is for }P(r).



    . . \begin{array}{cccccc}P(1)&=&\dfrac{n}{n}&=&1 \\ \\[-3mm] P(2)&=&\dfrac{n-1}{n} \\ \\[-3mm] P(3) &=& \dfrac{n-2}{n} \\ & \vdots \\ P(r) &=& \boxed{\dfrac{n-(r-1)}{n}} \end{array}

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