# Rolling a die

1. Apr 18, 2012

### cournot

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. Apr 18, 2012

### HallsofIvy

Staff Emeritus
What do you mean by "the pattern 1 2"?

3. Apr 18, 2012

### Ray Vickson

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