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I have a function that gives the probability of Y=1 given X i.e P(Y=1|X)=f(wX). This function is dependent on variables w and X and I have to give the range of w such that the following condition is satisfied:

There are two points namely X1 and X2. The question is "give the range of w such that X1 is more likely to have a positive sentiment than X2".

My interpretation is that "X1 must have a positive sentiment and X2 will not" since the question can be simply compared with "X1 is more likely to have chocolate than X2" which eases my understanding.

As a result, P(Y=1|X1) > 0.5 and P(Y=1|X2) < 0.5

However, the solution that my faculty has stated is P(Y=1|X1)>P(Y=1|X2) which conforms to the statement "X1 is likely to have more positive sentiment than X2".

Ultimately it boils down to two things:

1- What is the meaning of the statement: "X1 is more likely to have a positive sentiment than X2".

2- Is it different from "X1 is likely to have more positive sentiment than X2"

Kindly elaborate your thought process and help me find errors in any of the two approaches.

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I agree with your faculty. First, there is no indication that the probability of positive sentiment given X, P(Y=1|X), needs to above (or below) 0.5 for either case. For example, if P(Y=1|X1)=0.3 and P(Y=1|X2)=0.2, then using 0.5 as a cutoff point will not work. For this reason, any arbitrary cut off should not be used.

Second, in your interpretation that P(Y=1|X1)>P(Y=1|X2) means "X1 is likely to have more positive sentiment than X2" you are conflating the probability of positive sentiment with the amount of positive sentiment. Y appears to be a binary variable in this case, meaning that sentiment is either positive or it is not. What you are measuring with P(Y=1|X) is the probability that there is positive sentiment given X, not the amount of positive sentiment.

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