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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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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?

The Better Bet

Stage: 4 Challenge Level: Challenge Level:1
First the coins :

Imagine the four coins in a line, for example T H H H.
How many arrangements like this are there?
How many would win the $£3$?

If you played the game a specified number of times how much do you think you might have as profit or loss?
How much would that be on average per game?

Now the dice :

You could get $3$ sixes, what else could you get?
If you got $3$ sixes, how much would you win?
How often might you expect that to happen?
What about an overall expected profit or loss in the long run?