Expected number of runs

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  • #1
binjip
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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.
 
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  • #2
binjip said:

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.

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
 

What is the definition of expected number of runs?

The expected number of runs is a statistical concept that represents the average number of consecutive successes or failures in a sequence of independent events. It is calculated by multiplying the probability of success by the number of trials.

How is the expected number of runs calculated?

The expected number of runs is calculated by multiplying the probability of success by the number of trials. For example, if the probability of success is 0.5 and there are 10 trials, the expected number of runs would be 0.5 x 10 = 5.

What is the significance of expected number of runs in statistics?

The expected number of runs is significant in statistics because it helps to predict the likelihood of a specific outcome in a sequence of events. It also provides a measure of the variability or randomness in the data.

How does the expected number of runs differ from the actual number of runs?

The expected number of runs is a theoretical value based on the probability of success and the number of trials, while the actual number of runs is the real value observed in a given sequence of events. The two values may be different due to chance or other factors.

Can the expected number of runs be used to make predictions?

Yes, the expected number of runs can be used to make predictions about the likelihood of a specific outcome in a sequence of events. However, it should be noted that the expected number of runs is a theoretical value and may not always align with the actual number of runs in a given scenario.

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