1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Rolling a die

  1. Apr 18, 2012 #1
    1. The problem statement, all variables and given/known data
    a fair die is tossed n times
    let X be the number of times that the pattern 1 2 is observed
    (a) E(X) (b) VAR(X)

    (it's a question from a past exam)
    2. Relevant equations



    3. The attempt at a solution

    E(X)=E(E(X|first=1)+E(X|first ≠ 1))
    let E(X)=f(n)
    f(n)=((1+f(n-2))*(1/6)+(5/6)*f(n-1))*(1/6)+f(n-1)*(5/6)
    I think I'm setting up the equation wrong since I got something other than 12/(6^3) as I plug in n=3.
    However, even if I set it up correctly, I still have no idea how to solve this equation....
    Any help would be greatly appreciated.
     
  2. jcsd
  3. Apr 18, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    What do you mean by "the pattern 1 2"?
     
  4. Apr 18, 2012 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The following suggestions may be helpful (if you know something about Markov chains) or incomprehensible if you don't know Markov chains. Anyway, you can reduce it to a standard expected-cost calculation in a 3-state Markov chain. The three states are (1) state 1 (start, or start again); (2) state 2 (have tossed '1'); and (3) state 3 (have just achieved pattern '12'). We are in state 1 to start (before tossing). From state 1 we go go state 1 with probability 5/6 and to state 2 with probability 1/6. Frorm state 2 we go to state 1 with probability 4/6 (if we toss 3,4,5,6), to state 1 with probability 1/6 (if we toss 1, so throw out the previous '1' result and start another possible pattern), and to state 3 with probability 1/6 (if we toss '2'). From state 3 we go to state 1 with probability 5/6 and to state 2 with probability 1/6.

    Your random variable X is the number of times we visit state 3 in N tosses. We can regard this as an expected cost problem, where we pay $1 each time we hit state 3, and we are interested in the mean and variance of N-period cost. There are standard formulas that can be applied; these can be found in many sources dealing with Markov chains, but will not usually be found in introductory probability textbooks, for example.

    RGV
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook