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.
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?
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?
Here's how this bet works:
It involves nothing more than just tossing an ordinary coin until the first Head appears.
If it's a Head straight-off the prize is £2 and the game's over.
If instead it's a Tail the coin is tossed again.
The prize is now £4 for a Head if it shows, and the game will then be over.
But if it's a Tail, as before, the game continues.
The prize paid for the Head that finishes the game doubles each time a Tail turns up instead.
Not a bad game - you could get a very big prize.
On the other hand you may not get to the big prizes very often.
How many Tails would need to come up, before the winning Head, to win a million pounds or more ?
And how often would you expect to see that ?
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