Actually, that's incorrect and the contestant (I) was right. House of York first (under King Edward IV) defeated House of Lancaster (King Henry VI and subsequently, his son, Edward of Westminster). After Edward's death, however, his brother King Richard III (another York) seized the throne. He was ultimately defeated at Bosworth Field, the final battle of the Wars of the Roses, by Henry TUDOR (Henry VII), the Lancastrian successor to Henry VI via his mother, Catherine of Valois's remarriage to Owen Tudor (Henry Tudor's grandfather). Instead of keeping the Lancaster title and emblem, Henry VII chose to found a new house, the House of Tudor.
As you can see, I'm a passionate student of British history. That's why, when I got that question, I wanted to say the Tudor branch of Lancaster, and worded it Tudor or Lancaster just to be safe (no pause in between), because I did not want to be too specific. I guarantee you that had I just answered Tudor, I would've either gotten the credit then or after the break after they fact-checked.
Anyways, hope that helps clear that up. Certainly not defending all my answers, but feel very strongly about this one and thought you'd welcome the info.