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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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Balls and Bags

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

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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?

Fixing the Odds

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. Your friend chooses a bag at random and then chooses a ball at random from that bag. How should you distribute the balls between the two bags so as to make the probability that your friend will choose a red ball as small as possible and what will the probability be in that case?

How should you distribute the balls so as to make the probability of choosing a red ball as large as possible and what will the probability be in that case?

What happens if you have two bags, a hundred red balls and a hundred white balls?