Theoretical probability

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

Two bags contain different numbers of red and blue marbles. A marble is removed from one of the bags. The marble is blue. What is the probability that it was removed from bag A?

Two brothers belong to a club with 10 members. Four are selected for a match. Find the probability that both brothers are selected.

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.