Conditional probability and entropy: How do I interpret given data? As the title explains, I never

2nalfq8 2022-07-16 Answered
Conditional probability and entropy: How do I interpret given data?
As the title explains, I never could understand probabilities. It's one of those things that how much I try, I can't quite understand.
I have to do one homework exercise about entropy and I'm given a set of probabilities. I know how to calculate entropy but I don't know how to interpret the given data.
The alphabet is S={1,2} and the conditional probabilities are P(1|1)=0.8 P(2|1)=0.2 P(1|2)=0.6 P(2|2)=0.4 and P(1,2)=P(2,1)I've created this table (don't know if it is right or not):
| X = 1 | X = 2
Y = 1 | 0,8 | 0,2
Y = 2 | 0,6 | 0,4
I know that I need the probability of 1 and 2 to calculate the entropy. To get the probability of 1 is like this?
P(1)=P(1,2)P(2|1)
If so, How can I get P(1,2)?
I know that P(1)=0.75 and P(2)=0.25 but I don't understand how to get to this result
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Answers (1)

Zackery Harvey
Answered 2022-07-17 Author has 21 answers
here, the explicit solution: You have,
P ( 1 ) = P ( 1 , 2 ) P ( 2 | 1 )
and
P ( 2 ) = P ( 2 , 1 ) P ( 1 | 2 )
Because of
P(2,1)=P(1,2)
we get:
P ( 1 ) P ( 2 | 1 ) = P ( 2 ) P ( 1 | 2 )
P ( 1 ) = 3 P ( 2 )
Furthermore, you have P(1)+P(2)=1 and hence
4 P ( 2 ) = 1
Hence
P ( 2 ) = 0.25

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