A brainteaser I don't quite understand

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1

Hello, yesterday a professor showed us a problem in logic. People seemed to understand it, but I am lost. Can someone explain it to me?

Three Masters of Logic wanted to find out who was the wisest amongst them. So 
they turned to their Grand Master, asking to resolve their dispute. “Easy,” the old 
sage said. “I will blindfold you and paint either red, or blue dot on each man's 
forehead. When I take your blindfolds off, if you see at least one red dot, raise 
your hand. The one, who guesses the color of the dot on his forehead first, wins.
”And so it was said, and so it was done. The Grand Master blindfolded the three 
contestants and painted red dots on every one. When he took their blindfolds off, 
all three men raised their hands as the rules required, and sat in silence 
pondering. Finally, one of them said: “I have a red dot on my forehead.”

A: How did the winner guess?

I understand that the first time the winner reacted, because both opponents have been silent, so they both had to see two red dots. But the second part is a mystery to me.

After losing the “Spot on the Forehead” contest, the two defeated Puzzle Masters 
complained that the winner had made a slight pause before raising his hand, thus 
derailing their deductive reasoning train of thought. And so the Grand Master vowed 
to set up a truly fair test to reveal the best logician amongst them. He showed the 
three men 5 hats - two white and three black. Then he turned off the lights in the 
room and put a hat on each Puzzle Master's head. After that the old sage hid the 
remaining two hats, but before he could turn the lights on, one of the Masters (as 
chance would have it, the winner of the previous contest) announced the color of 
his hat. And he was right once again.

B: What color was his hat? What could have been his reasoning?

I presume this has nothing to do with probability, because then he could just have had luck. Is it because he suspected the Grand Master to repeat the trick and give them all black hats as there were only three white ones?

Well it also had to be this way, because if he gave a white hat to someone, then that person would automatically loose, which is unfair.

2

I guess the professor wanted to show you two interesting aspects of logic:

  1. The ability to get information about an entire dataset from an incomplete dataset
  2. The ability to apply gathered information from solved problems to new problems

Both have applications in big data, machine learning, AI etc.

3

Im guessing the masters answer to the problems was the same, everyones equals or what not. Everyone had red dots, and everyone got black hats. Does'nt matter what the marker is.

4

It has to do with the wording; the second test is "truly fair", other rather is supposed to offer an equal chance of everyone winning. However, with two white hats and three black hats, the only fair/equal way for any of them to win is if all them had black hats.

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