### 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.

### Marbles and Bags

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?

### 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?

# Bet You a Million

##### Stage: 4 Challenge Level:

Good straight-forward logic from Thomas in Dalton, New York

The number of throws to get a million pounds would be at least 20 since 2^20 is 1,048,576.

This would require throwing at least 19 tails sequentially, which has a probability of 1/ 2^19 or 1/524,288, or approximately 0.000190735 percent.

The amount you should pay to play the game is less than or equal to the expected value of the winnings.

After the first turn the expected winnings are 1/2*2 = 1 pound.

After two turns the expected winnings are 1 + 1/4*4 = 2 pounds.

After n turns the expected value is n pounds; therefore, one would pay any amount of money, assuming that there's no Gambler's ruin.