Three human subjects underwent an experiment. They were each given one letter that could be found in the following words:

The, fit, lid, bet, and big.

If all three put their letters together, they would be able to spell one of the words above. However, the letter they were given could not be shared with any of the other four. They were all in the same room when a woman came through the door and sat down in front of them. She asked the first patient if he knew the what the word was that could be spelled using all three of the letters. He said yes. She then asked the other two patients and they both said yes.

What was the word and how did all three know having only one letter and not knowing the other two?