[nomadking]

The 1st round is immaterial, as you still don't know whether your 1st choice was right or wrong. There is a 50/50 chance of selecting the right one. Doesn't matter whether you stick with or swap your original choice.

There is no win or lose event for the 1st round, so you can't multiply the two probabilities together. Doesn't matter which one of the 3 you pick in the first round, as you are starting again with just 2 choices.

If it was 2 rounds with 2 different people separately and independently, making the selection and both had to be right, then yes you can multiply the probabilities together to get a 1 in 6 chance of both people being right.