He didn’t know she knew the answer...if she wou,d have et two dollars he wou,d have won.
Most people in the lead bet enough to make sure that, if they get the answer right, they will win (i.e., enough to end up more than double the second place person). Knowing that, the second place person should bet such that, if the 1st place person misses the answer, they will have more money (regardless of their answer, right or wrong). Which is what James did, with the additional knowledge that he had to remain above double the third place person, whether he was right or wrong.