Logistic regression and classification: Adjusting or removing decision boundaries I'm taking Andrew Ng's Machine Learning Course. In the lesson on classification algorithms, he presents the logit function ($\frac{1}{1+e^{-x}}$) and the way it converts parameterized functions to probabilities. He marks the "decision boundary" at .5, noting that the logit function cross .5 at $x=0$. If the value of the logistic regression function is .5 or greater, the result is given as 1. If it's less than .5, it's given as 0.
This sounds arbitrary. In real-world applications, is it common to adjust the decision boundary, or disregard the boundary and list only the probabilities? 


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*Scenario 1 Adjusting the decision boundary: In the logit function, when x is -.3, y is .4256. Because y is less than .5, the predicted value is 0. If the lump in my patient's breast has a 42.56% probability of being malignant, I don't want that rounded down to 0%.
I would adjust the decision boundary from .5 to maybe .01. 1% or less probability is the threshold where I'd tell my patient their lump is nothing to worry about.

*Scenario 2 Removing the decision boundary: My team of social workers helps teens in my city. My classification parameters predict that 15% of the teen population is at high risk of committing a crime. That is, it predicts a positive result for 15% of P. My team is small, however, and can only reach 3% of the teen population. 
Knowing that the 15% prediction includes teens at 51% probability along with teens at 99% probability, I disregard the binary predictions entirely and order teens based on their probability, thus reaching the most at-risk teens first.
Are these scenarios—where you adjust decision boundaries and/or list probabilities alone—common in real-world usage?
 A: *

*Logistic regression can be viewed as a model output probabilities, not just a classifier.

*As @Matthew Drury mentioned, adjusting the boundary is used very frequently in real world to make trade-off between true positive and false positive rates. Details see [ROC curve] for details (https://en.wikipedia.org/wiki/Receiver_operating_characteristic)



In addition to optimizing the logistic loss, which is the view from the machine learning perspective. There are very rich literature in statistics about logistic regression can answer your question perfectly and give more interpretation of the probability produced from the model. 
These literatures also establish the relationship between max likelihood and binomial assumptions of logistic regression. (Note, not any number between 0 and 1 is a probability number. When we talk about probability we also want to think about the distribution.) The link function (sigmoid, $\text{logit}^{-1}$) is carefully designed with rigorous math behind.   
I would strongly recommend you to re-learn logistic regression from statistics literature in addition to machine learning. 
Here are some starting resources:
FAQ: HOW DO I INTERPRET ODDS RATIOS IN LOGISTIC REGRESSION?
LOGIT REGRESSION | R DATA ANALYSIS EXAMPLE
And some related topics in CV
What is the difference between linear regression and logistic regression?
Is there i.i.d. assumption on logistic regression?
