Yes. Gambling income in the state where it was received, and the fact that Missouri withheld tax in and of itself means that you will file a Missouri nonresident return to determine how much of those withholdings will stay with Missouri or if you will receive a refund of some of them. When you prepare your state returns, you will want to prepare the Missouri nonresident return first.
That is because your resident state also taxes your income. In order to reduce the effect of double-taxation, your resident state will give you a credit for the taxes you will pay to Missouri on the gambling income. However, you will still have to file tax returns for both states (unless you live in a state without income tax, in which case you only file for Missouri).
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Yes
No
