Probability of θ-Ω-θ-Ω-θ Sequence

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Discussion Overview

The discussion revolves around calculating the probability of a specific sequence of characters (Ω, θ, β) with given probabilities, focusing on a sequence of six characters where the first and last characters are θ. The scope includes mathematical reasoning and exploratory approaches to probability calculation.

Discussion Character

  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant suggests constructing all possible sequences that fit the pattern θ_ _ _ _ θ to determine the probabilities.
  • Another participant proposes an alternative approach by calculating the probability that neither the first nor the last characters is θ, considering different sequences that could occur in that case.
  • There is a suggestion that the probability could be simplified to (0.3)^2 if the middle characters do not affect the outcome, although this is presented as a guess.
  • One participant acknowledges a mistake regarding the sum of probabilities and expresses uncertainty about how to approach the problem without considering all cases separately.
  • Another participant states that the probability of the sequence could be expressed as (p(θ))^2.
  • One participant admits to misreading the problem and expresses a need to revisit it for clarity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to calculate the probability, with multiple competing approaches and uncertainties expressed throughout the discussion.

Contextual Notes

Some participants note limitations in their understanding of the problem and the need to consider various cases, indicating that assumptions about the independence of events may not hold.

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Given three characters that have a specific probability assigned to each of them:
Ω: .2
θ: .3
β: .4

What is the probability of having, say a sequence of 6 of these characters where the first and last character is a θ?
 
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First of all: I'm not crazy, Institutions! :)

Secondly, this is a sort-of rote matter of constructing all sequences:

θ_ _ _ _ θ

and then figuring out the probabilities for all of the sequences of this type. I can't

think of a shortcut.
 
Bacle2 said:
First of all: I'm not crazy, Institutions! :)

Secondly, this is a sort-of rote matter of constructing all sequences:

θ_ _ _ _ θ

and then figuring out the probabilities for all of the sequences of this type. I can't

think of a shortcut.

glad you got the reference! But not so glad I have to construct a tree to solve this problem...
 
There may be an alternative: find the probability that neither the first , nor the last

elements is a θ, so you find the probability of having one of the 4 sequences:

β_ _ _ _β , β_ _ _ _Ω , Ω _ _ _ _β , Ω _ _ _ _Ω , or maybe less , depending if you

consider the order does not matter, and then the 2nd and 3rd sequences are the same.
 
is it possible then that the answer is (0.3)^2 because the letters in the middle do not really matter? This is just me guessing but if the stuff in the middle doesn't matter then we can simply treat the whole problem as if it were a sequence of two?
 
Oh and I made a mistake..those probabilities should add up to 1.
 
Actually, my bad; I was assuming every event had the same probability. I really

can't see any option other than considering all the cases separately. Sorry.
 
The probability of the sequence is simply [itex](p(\theta))^2[/itex].
 
Ouch, I think I misread the problem. Sorry.; let me read it again more

carefully.
 

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