Some Interesting Probability Questions

I have collected these problems from various sources, and claim that I have not formulated them on my own.

1. A and B are playing a game where A has n+1 coins, B has n coins, and they each flip all of their coins. What is the probability that A will have more heads than B? Assume all the coins are fair.
   Answer

   1/2. Hint : Try to start with n = 1.
2. A person flips an unbiased coin over and over again. Player A looks for the sequence HHT and player B looks for the sequence HTT. What is the probability that player A encounters their sequence first?
   Answer

   2/3.
3. You are given a choice of three doors by an Angel. You can choose only one of the doors among the three. Out of these three doors, two contain nothing and one has a jackpot. After you choose one of the doors, the angel reveals one of the other two doors behind which there is nothing. The angel gives you an opportunity to change the door or you can stick with your chosen door. You don’t know behind which door we have nothing. Should you switch?
   Answer

   YES. This is the infamous Monty-Hall Problem .