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Gambling at Monte Carlo

A man went to Monte Carlo to try and make his fortune. Is his strategy a winning one?

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Coin Tossing Games

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?

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Win or Lose?

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?

Marbles and Bags

Age 14 to 16 Challenge Level:

Assume that removing a ball from bag A or bag B is equally likely, the probability of choosing a blue ball from bag A is 3/10 and the probability of choosing a blue ball from bag B is 4/14 so the probability of choosing a blue ball is 3/10 + 4/14 = 41/70.
Theoretically in 70 trials, 41 of the outcomes would give a blue ball and 21 of these would of been drawn from bag A . Given that the ball drawn was blue the probability that it came from bag A is 21/41.

The only correct solution to date came from Matthew (Smithdon High School)