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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

Age 14 to 16 Challenge Level:
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?