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Gambling at Monte Carlo

A man went to Monte Carlo to try and make his fortune. Whilst he was there he had an opportunity to bet on the outcome of rolling dice. He was offered the same odds for each of the following outcomes: At least 1 six with 6 dice. At least 2 sixes with 12 dice. At least 3 sixes with 18 dice.

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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

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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)