I don't think it matters:
1) If you are #1 candidate, most likely you will receive offer.
2) If you are not #1 cadidate, most like you will not receive offer unless #1 candidate turned down offer.
Actually, there was only one time that thank-you letter helped me to get offer: I answered a question right, but the interviewer thought I answered wrong, so I sent a thank-you letter and proved I was right. At the same day, I received the offer.