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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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Fixing the Odds

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

Win or Lose?

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

The gambler will have less money than he started with.

Suppose the amount of money before a game is $m$, then:

$m \to 3m/2$ for a win and $m\to m/2$ after losing a game.

Values of n Amount after 2n games: n wins, n losses
1 $3m/4$
2 $m \times 1/2 \times3/2 \times1/2 \times3/2 = (3/4)^2 m$
3 $m \times1/2 \times3/2 \times1/2 \times3/2 \times1/2 \times3/2 = (3/4)^3 m$

After $n$ wins and $n$ losses he will have $(3/4)^n$ times the money he started with, irrespective of the order in which his wins and losses occur. Eventually he will run out of money as what he has left will be smaller than the smallest coin in circulation.

The diagram was suggested by Roderick and Michael of Simon Langton Boys' Grammar School Canterbury who pointed out that if the gambler went on indefinitely he would, in theory, end up with an infinitely small amount which would be represented by nothing.