1. The problem statement, all variables and given/known data Hi, Suppose we have a die with 3 colors on it. 4 sides are blue => P(Z=Blue) = 2/3 1 side is green => P(Z=Green) = 1/6 1 side is red => P(Z=Red) = 1/6 I throw it 20 times and have Z=(Z_{1,..., Z20}). Now what is the expected number of "runs"? Run is defined as the number of times the color changes, or equivalently, as the number of consistent blocks of a color. For example: string "bbgrg" has 4 runs ( |bb|, |g|, |r|, |g| ) 2. Relevant equations 3. The attempt at a solution Attempt #1: Change the representation of the sequence from "bbgrg" into a sequence of 1 and 0. One being a new color block (a success), 0 being just another ball of the previous color. "bbgrg" becomes 10111. In other words, P(X_{i}=1), if {Z_{i} != Z_{i+1}}. This is, however, only a restatement of the problem and doesn't solve the initial problem: how many "1" do I have in 20 throws? Attempt #2: The number of throws before a given color occurs is geometrically distributed (Geo(p)). Thus: E(number of throws until blue occurs) = 1/P(Z=Blue) = 3/2 E(number of throws until green occurs) = 1/P(Z=Green) = 6 E(number of throws until red occurs) = 1/P(Z=Red) = 6 I also know E(# Blue) = n * P(Z=Blue) = 20*2/3 = 40/3 E(# Green) = E(# Red) = 20/6 I could maybe use those 2 pieces of information but I can't see how. Any comments are welcomed. Thank you for help. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
I would do it by an iterative method. If B(n) = expected number of runs in n tosses, given the first toss is Blue, and G(n), R(n) are defined similarly, I would get the answer in terms of B(20), G(20) and R(20). Then I would get recursions for B(n), G(n) and R(n) by noting how B(n) is related to B(n-1), G(n-1) and R(n-1) by looking at the next colour, etc. RGV