- #1
binjip
- 6
- 0
Homework Statement
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=(Z1,..., 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| )
Homework Equations
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(Xi=1), if {Zi != Zi+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.