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How to determine if my roulette wins were due to luck or due to skill?

Someone told me that question above is ungrammatical and should be changed to:

How can I determine if my roulette wins were due to luck or due to skill?

Is my original sentence incorrect? If so, why?

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They were correct, starting a sentence with "How to" is invalid in English.

"How to" is a fragment of a complete sentence such as "I would like to learn how to sing", but cannot be used on its own in this way.

The sentence:

How to determine if my roulette wins were due to luck or due to skill?

does not have any subject or verb which makes it incorrect.

The sentence:

How can I determine if my roulette wins were due to luck or due to skill?

is fully formed, containing both the subject, I, and the modal verb can.

This is a common error that non-English speakers make frequently, and is a giveaway that the speaker is not yet fluent.

Unrelated to your question, the ending of your sentence

"...due to luck or due to skill?"

is somewhat verbose, and a native speaker would likely say this as:

"...due to luck or skill?"

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