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I just wanted to confirm if this was correct :)

For logistic regressions, do the outputs in the training set are actually no different than the outputs that we get as predictions? Is the only reason we have binary outputs as part of the classification for logistic regression is that we know it to be true or not[a student gets admitted or not; the cancer is benign or not].

Technically, could we still have the output as any real value between 0 and 1 in the training set -- e.g., we know %60 that the cancer is benign or %60 that the student will get admitted?

Just to illustrate, let's say we have the outputs as 0's and 1's in a scenario as

34.62365962451697, 78.0246928153624, 0*

*where the first two columns are test scores and the third one is the college admission decision

Would it be okay if we change the 0 to 0.6 and then run logistic regression [assuming the student's chance of admission is 60%]:

34.62365962451697, 78.0246928153624, 0.6

Your answer will be much appreciated!

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Yes you can. For example, R enables you to use logistic regression with three kinds of data

For the binomial and quasibinomial families the response can be specified in one of three ways:

  1. As a factor: ‘success’ is interpreted as the factor not having the first level (and hence usually of having the second level).

  2. As a numerical vector with values between 0 and 1, interpreted as the proportion of successful cases (with the total number of cases given by the weights).

  3. As a two-column integer matrix: the first column gives the number of successes and the second the number of failures.

All the three ways of storing data are equivalent, so it is just a matter of interface. Notice however that for second way you need not only the proportion, but also the sample size. This also may not be possible if you use other software that does not allow for providing the data in such form.

However, if you want to model outcomes between 0 and 1 that are not proportions with sample size given, you should rather consider other approaches e.g. beta regression as noticed by @Alex.

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Traditional logistic regression requires that the response variable be zero or one (or an integer, in the case of binomial regression).

If you somehow know the student's admission chance (how on earth do you know that?) then you need to use a different distribution, one which takes values on $[0,1]$ rather than $\{0,1\}$. The usual distribution to use in that case is a beta distribution, but there are others.

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