The simplest definition of a holder in due course is as follows: (i) a person who receives a check in good faith and as an exchange for value and (ii) has no suspicion that the check has a prior claim by another party and (iii) has no knowledge that the...
The simplest definition of a holder in due course is as follows: (i) a person who receives a check in good faith and as an exchange for value and (ii) has no suspicion that the check has a prior claim by another party and (iii) has no knowledge that the instrument was previously dishonored qualifies as a holder in due course. The holder in due course statute can be found in the Uniform Commercial Code, Section 3-302.
Based on the facts in your example, Carol has received the check from the teller as payment for a gambling debt, so the check has been given to her in exchange for value (that is, the gambling debt). Carol also has no suspicion, based on the presentation of the check, that the bank for whom the teller works has a prior claim to the check--in other words, she has no reason to question the legitimacy of the check or that it might have been previously dishonored. Under Section 3-302(a)(i) of the UCC, the instrument cannot appear to be "irregular or incomplete as to call into question its authenticity." So, the bank might argue that Carol should have been suspicious of the check's origin because it has been signed by a third party--that is, not the teller who owes the debt--but the teller could easily allay Carol's suspicion by lying convincingly that the check represents a debt owed to the teller by the signatory of the check. In other words, there is nothing inherent in the instrument itself that would cause Carol to doubt its authenticity or its negotiability.
As the holder in due course, Carol has the right to endorse the check and receive the funds. Unless the bank, which has been defrauded by its teller, can prove that Carol somehow should have known that the instrument is not negotiable, the bank's recourse is against the teller, not Carol.